GAP (MonoidalCategories) - Chapter 1: Monoidal Categories

Goto Chapter: Top 1 2 3 4 5 6 Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

[MathJax on]

1 Monoidal Categories
1.1 Monoidal Categories
1.1-1 TensorProductOnMorphisms
1.1-2 TensorProductOnMorphismsWithGivenTensorProducts
1.1-3 AssociatorRightToLeft
1.1-4 AssociatorRightToLeftWithGivenTensorProducts
1.1-5 AssociatorLeftToRight
1.1-6 AssociatorLeftToRightWithGivenTensorProducts
1.1-7 LeftUnitor
1.1-8 LeftUnitorWithGivenTensorProduct
1.1-9 LeftUnitorInverse
1.1-10 LeftUnitorInverseWithGivenTensorProduct
1.1-11 RightUnitor
1.1-12 RightUnitorWithGivenTensorProduct
1.1-13 RightUnitorInverse
1.1-14 RightUnitorInverseWithGivenTensorProduct
1.1-15 TensorProductOnObjects
1.1-16 AddTensorProductOnObjects
1.1-17 TensorUnit
1.1-18 AddTensorUnit
1.2 Additive Monoidal Categories
1.2-1 LeftDistributivityExpanding
1.2-2 LeftDistributivityExpandingWithGivenObjects
1.2-3 LeftDistributivityFactoring
1.2-4 LeftDistributivityFactoringWithGivenObjects
1.2-5 RightDistributivityExpanding
1.2-6 RightDistributivityExpandingWithGivenObjects
1.2-7 RightDistributivityFactoring
1.2-8 RightDistributivityFactoringWithGivenObjects
1.3 Braided Monoidal Categories
1.3-1 Braiding
1.3-2 BraidingWithGivenTensorProducts
1.3-3 BraidingInverse
1.3-4 BraidingInverseWithGivenTensorProducts
1.4 Symmetric Monoidal Categories
1.5 Left Closed Monoidal Categories
1.5-1 LeftInternalHomOnObjects
1.5-2 LeftInternalHomOnMorphisms
1.5-3 LeftInternalHomOnMorphismsWithGivenLeftInternalHoms
1.5-4 LeftClosedMonoidalEvaluationMorphism
1.5-5 LeftClosedMonoidalEvaluationMorphismWithGivenSource
1.5-6 LeftClosedMonoidalCoevaluationMorphism
1.5-7 LeftClosedMonoidalCoevaluationMorphismWithGivenRange
1.5-8 TensorProductToLeftInternalHomAdjunctMorphism
1.5-9 TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom
1.5-10 LeftInternalHomToTensorProductAdjunctMorphism
1.5-11 LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct
1.5-12 LeftClosedMonoidalPreComposeMorphism
1.5-13 LeftClosedMonoidalPreComposeMorphismWithGivenObjects
1.5-14 LeftClosedMonoidalPostComposeMorphism
1.5-15 LeftClosedMonoidalPostComposeMorphismWithGivenObjects
1.5-16 LeftDualOnObjects
1.5-17 LeftDualOnMorphisms
1.5-18 LeftDualOnMorphismsWithGivenLeftDuals
1.5-19 LeftClosedMonoidalEvaluationForLeftDual
1.5-20 LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct
1.5-21 MorphismToLeftBidual
1.5-22 MorphismToLeftBidualWithGivenLeftBidual
1.5-23 TensorProductLeftInternalHomCompatibilityMorphism
1.5-24 TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects
1.5-25 TensorProductLeftDualityCompatibilityMorphism
1.5-26 TensorProductLeftDualityCompatibilityMorphismWithGivenObjects
1.5-27 MorphismFromTensorProductToLeftInternalHom
1.5-28 MorphismFromTensorProductToLeftInternalHomWithGivenObjects
1.5-29 IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit
1.5-30 IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject
1.5-31 UniversalPropertyOfLeftDual
1.5-32 LeftClosedMonoidalLambdaIntroduction
1.5-33 LeftClosedMonoidalLambdaElimination
1.5-34 IsomorphismFromObjectToLeftInternalHom
1.5-35 IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom
1.5-36 IsomorphismFromLeftInternalHomToObject
1.5-37 IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom
1.6 Closed Monoidal Categories
1.6-1 InternalHomOnObjects
1.6-2 InternalHomOnMorphisms
1.6-3 InternalHomOnMorphismsWithGivenInternalHoms
1.6-4 ClosedMonoidalRightEvaluationMorphism
1.6-5 ClosedMonoidalRightEvaluationMorphismWithGivenSource
1.6-6 ClosedMonoidalRightCoevaluationMorphism
1.6-7 ClosedMonoidalRightCoevaluationMorphismWithGivenRange
1.6-8 TensorProductToInternalHomRightAdjunctMorphism
1.6-9 TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom
1.6-10 TensorProductToInternalHomRightAdjunctionIsomorphism
1.6-11 TensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects
1.6-12 InternalHomToTensorProductRightAdjunctMorphism
1.6-13 InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct
1.6-14 InternalHomToTensorProductRightAdjunctionIsomorphism
1.6-15 InternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects
1.6-16 ClosedMonoidalLeftEvaluationMorphism
1.6-17 ClosedMonoidalLeftEvaluationMorphismWithGivenSource
1.6-18 ClosedMonoidalLeftCoevaluationMorphism
1.6-19 ClosedMonoidalLeftCoevaluationMorphismWithGivenRange
1.6-20 TensorProductToInternalHomLeftAdjunctMorphism
1.6-21 TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom
1.6-22 TensorProductToInternalHomLeftAdjunctionIsomorphism
1.6-23 TensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects
1.6-24 InternalHomToTensorProductLeftAdjunctMorphism
1.6-25 InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct
1.6-26 InternalHomToTensorProductLeftAdjunctionIsomorphism
1.6-27 InternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects
1.6-28 MonoidalPreComposeMorphism
1.6-29 MonoidalPreComposeMorphismWithGivenObjects
1.6-30 MonoidalPostComposeMorphism
1.6-31 MonoidalPostComposeMorphismWithGivenObjects
1.6-32 DualOnObjects
1.6-33 DualOnMorphisms
1.6-34 DualOnMorphismsWithGivenDuals
1.6-35 EvaluationForDual
1.6-36 EvaluationForDualWithGivenTensorProduct
1.6-37 MorphismToBidual
1.6-38 MorphismToBidualWithGivenBidual
1.6-39 TensorProductInternalHomCompatibilityMorphism
1.6-40 TensorProductInternalHomCompatibilityMorphismWithGivenObjects
1.6-41 TensorProductDualityCompatibilityMorphism
1.6-42 TensorProductDualityCompatibilityMorphismWithGivenObjects
1.6-43 MorphismFromTensorProductToInternalHom
1.6-44 MorphismFromTensorProductToInternalHomWithGivenObjects
1.6-45 IsomorphismFromDualObjectToInternalHomIntoTensorUnit
1.6-46 IsomorphismFromInternalHomIntoTensorUnitToDualObject
1.6-47 UniversalPropertyOfDual
1.6-48 LambdaIntroduction
1.6-49 LambdaElimination
1.6-50 IsomorphismFromObjectToInternalHom
1.6-51 IsomorphismFromObjectToInternalHomWithGivenInternalHom
1.6-52 IsomorphismFromInternalHomToObject
1.6-53 IsomorphismFromInternalHomToObjectWithGivenInternalHom
1.7 Left Coclosed Monoidal Categories
1.7-1 LeftInternalCoHomOnObjects
1.7-2 LeftInternalCoHomOnMorphisms
1.7-3 LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms
1.7-4 LeftCoclosedMonoidalEvaluationMorphism
1.7-5 LeftCoclosedMonoidalEvaluationMorphismWithGivenRange
1.7-6 LeftCoclosedMonoidalCoevaluationMorphism
1.7-7 LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource
1.7-8 TensorProductToLeftInternalCoHomAdjunctMorphism
1.7-9 TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom
1.7-10 LeftInternalCoHomToTensorProductAdjunctMorphism
1.7-11 LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct
1.7-12 LeftCoclosedMonoidalPreCoComposeMorphism
1.7-13 LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects
1.7-14 LeftCoclosedMonoidalPostCoComposeMorphism
1.7-15 LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects
1.7-16 LeftCoDualOnObjects
1.7-17 LeftCoDualOnMorphisms
1.7-18 LeftCoDualOnMorphismsWithGivenLeftCoDuals
1.7-19 LeftCoclosedMonoidalEvaluationForLeftCoDual
1.7-20 LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct
1.7-21 MorphismFromLeftCoBidual
1.7-22 MorphismFromLeftCoBidualWithGivenLeftCoBidual
1.7-23 LeftInternalCoHomTensorProductCompatibilityMorphism
1.7-24 LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
1.7-25 LeftCoDualityTensorProductCompatibilityMorphism
1.7-26 LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects
1.7-27 MorphismFromLeftInternalCoHomToTensorProduct
1.7-28 MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects
1.7-29 IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit
1.7-30 IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject
1.7-31 UniversalPropertyOfLeftCoDual
1.7-32 LeftCoclosedMonoidalLambdaIntroduction
1.7-33 LeftCoclosedMonoidalLambdaElimination
1.7-34 IsomorphismFromObjectToLeftInternalCoHom
1.7-35 IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom
1.7-36 IsomorphismFromLeftInternalCoHomToObject
1.7-37 IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom
1.8 Coclosed Monoidal Categories
1.8-1 InternalCoHomOnObjects
1.8-2 InternalCoHomOnMorphisms
1.8-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms
1.8-4 CoclosedMonoidalRightEvaluationMorphism
1.8-5 CoclosedMonoidalRightEvaluationMorphismWithGivenRange
1.8-6 CoclosedMonoidalRightCoevaluationMorphism
1.8-7 CoclosedMonoidalRightCoevaluationMorphismWithGivenSource
1.8-8 TensorProductToInternalCoHomRightAdjunctMorphism
1.8-9 TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom
1.8-10 InternalCoHomToTensorProductRightAdjunctMorphism
1.8-11 InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct
1.8-12 CoclosedMonoidalLeftEvaluationMorphism
1.8-13 CoclosedMonoidalLeftEvaluationMorphismWithGivenRange
1.8-14 CoclosedMonoidalLeftCoevaluationMorphism
1.8-15 CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource
1.8-16 TensorProductToInternalCoHomLeftAdjunctMorphism
1.8-17 TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom
1.8-18 InternalCoHomToTensorProductLeftAdjunctMorphism
1.8-19 InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct
1.8-20 MonoidalPreCoComposeMorphism
1.8-21 MonoidalPreCoComposeMorphismWithGivenObjects
1.8-22 MonoidalPostCoComposeMorphism
1.8-23 MonoidalPostCoComposeMorphismWithGivenObjects
1.8-24 CoDualOnObjects
1.8-25 CoDualOnMorphisms
1.8-26 CoDualOnMorphismsWithGivenCoDuals
1.8-27 CoclosedEvaluationForCoDual
1.8-28 CoclosedEvaluationForCoDualWithGivenTensorProduct
1.8-29 MorphismFromCoBidual
1.8-30 MorphismFromCoBidualWithGivenCoBidual
1.8-31 InternalCoHomTensorProductCompatibilityMorphism
1.8-32 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
1.8-33 CoDualityTensorProductCompatibilityMorphism
1.8-34 CoDualityTensorProductCompatibilityMorphismWithGivenObjects
1.8-35 MorphismFromInternalCoHomToTensorProduct
1.8-36 MorphismFromInternalCoHomToTensorProductWithGivenObjects
1.8-37 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
1.8-38 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
1.8-39 UniversalPropertyOfCoDual
1.8-40 CoLambdaIntroduction
1.8-41 CoLambdaElimination
1.8-42 IsomorphismFromObjectToInternalCoHom
1.8-43 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
1.8-44 IsomorphismFromInternalCoHomToObject
1.8-45 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
1.9 Symmetric Closed Monoidal Categories
1.10 Symmetric Coclosed Monoidal Categories
1.11 Rigid Symmetric Closed Monoidal Categories
1.11-1 IsomorphismFromTensorProductWithDualObjectToInternalHom
1.11-2 IsomorphismFromInternalHomToTensorProductWithDualObject
1.11-3 MorphismFromInternalHomToTensorProduct
1.11-4 MorphismFromInternalHomToTensorProductWithGivenObjects
1.11-5 TensorProductInternalHomCompatibilityMorphismInverse
1.11-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
1.11-7 CoevaluationForDual
1.11-8 CoevaluationForDualWithGivenTensorProduct
1.11-9 TraceMap
1.11-10 RankMorphism
1.11-11 MorphismFromBidual
1.11-12 MorphismFromBidualWithGivenBidual
1.12 Rigid Symmetric Coclosed Monoidal Categories
1.12-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject
1.12-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
1.12-3 MorphismFromTensorProductToInternalCoHom
1.12-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects
1.12-5 InternalCoHomTensorProductCompatibilityMorphismInverse
1.12-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
1.12-7 CoclosedCoevaluationForCoDual
1.12-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct
1.12-9 CoTraceMap
1.12-10 CoRankMorphism
1.12-11 MorphismToCoBidual
1.12-12 MorphismToCoBidualWithGivenCoBidual
1.13 Convenience Methods
1.13-1 InternalHom
1.13-2 InternalCoHom
1.13-3 LeftInternalHom
1.13-4 LeftInternalCoHom
1.14 Add-methods
1.14-1 AddLeftDistributivityExpanding
1.14-2 AddLeftDistributivityExpandingWithGivenObjects
1.14-3 AddLeftDistributivityFactoring
1.14-4 AddLeftDistributivityFactoringWithGivenObjects
1.14-5 AddRightDistributivityExpanding
1.14-6 AddRightDistributivityExpandingWithGivenObjects
1.14-7 AddRightDistributivityFactoring
1.14-8 AddRightDistributivityFactoringWithGivenObjects
1.14-9 AddBraiding
1.14-10 AddBraidingInverse
1.14-11 AddBraidingInverseWithGivenTensorProducts
1.14-12 AddBraidingWithGivenTensorProducts
1.14-13 AddClosedMonoidalLeftCoevaluationMorphism
1.14-14 AddClosedMonoidalLeftCoevaluationMorphismWithGivenRange
1.14-15 AddClosedMonoidalLeftEvaluationMorphism
1.14-16 AddClosedMonoidalLeftEvaluationMorphismWithGivenSource
1.14-17 AddClosedMonoidalRightCoevaluationMorphism
1.14-18 AddClosedMonoidalRightCoevaluationMorphismWithGivenRange
1.14-19 AddClosedMonoidalRightEvaluationMorphism
1.14-20 AddClosedMonoidalRightEvaluationMorphismWithGivenSource
1.14-21 AddDualOnMorphisms
1.14-22 AddDualOnMorphismsWithGivenDuals
1.14-23 AddDualOnObjects
1.14-24 AddEvaluationForDual
1.14-25 AddEvaluationForDualWithGivenTensorProduct
1.14-26 AddInternalHomOnMorphisms
1.14-27 AddInternalHomOnMorphismsWithGivenInternalHoms
1.14-28 AddInternalHomOnObjects
1.14-29 AddInternalHomToTensorProductLeftAdjunctMorphism
1.14-30 AddInternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct
1.14-31 AddInternalHomToTensorProductLeftAdjunctionIsomorphism
1.14-32 AddInternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects
1.14-33 AddInternalHomToTensorProductRightAdjunctMorphism
1.14-34 AddInternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct
1.14-35 AddInternalHomToTensorProductRightAdjunctionIsomorphism
1.14-36 AddInternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects
1.14-37 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit
1.14-38 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject
1.14-39 AddIsomorphismFromInternalHomToObject
1.14-40 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom
1.14-41 AddIsomorphismFromObjectToInternalHom
1.14-42 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom
1.14-43 AddLambdaElimination
1.14-44 AddLambdaIntroduction
1.14-45 AddMonoidalPostComposeMorphism
1.14-46 AddMonoidalPostComposeMorphismWithGivenObjects
1.14-47 AddMonoidalPreComposeMorphism
1.14-48 AddMonoidalPreComposeMorphismWithGivenObjects
1.14-49 AddMorphismFromTensorProductToInternalHom
1.14-50 AddMorphismFromTensorProductToInternalHomWithGivenObjects
1.14-51 AddMorphismToBidual
1.14-52 AddMorphismToBidualWithGivenBidual
1.14-53 AddTensorProductDualityCompatibilityMorphism
1.14-54 AddTensorProductDualityCompatibilityMorphismWithGivenObjects
1.14-55 AddTensorProductInternalHomCompatibilityMorphism
1.14-56 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects
1.14-57 AddTensorProductToInternalHomLeftAdjunctMorphism
1.14-58 AddTensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom
1.14-59 AddTensorProductToInternalHomLeftAdjunctionIsomorphism
1.14-60 AddTensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects
1.14-61 AddTensorProductToInternalHomRightAdjunctMorphism
1.14-62 AddTensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom
1.14-63 AddTensorProductToInternalHomRightAdjunctionIsomorphism
1.14-64 AddTensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects
1.14-65 AddUniversalPropertyOfDual
1.14-66 AddCoDualOnMorphisms
1.14-67 AddCoDualOnMorphismsWithGivenCoDuals
1.14-68 AddCoDualOnObjects
1.14-69 AddCoDualityTensorProductCompatibilityMorphism
1.14-70 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects
1.14-71 AddCoLambdaElimination
1.14-72 AddCoLambdaIntroduction
1.14-73 AddCoclosedEvaluationForCoDual
1.14-74 AddCoclosedEvaluationForCoDualWithGivenTensorProduct
1.14-75 AddCoclosedMonoidalLeftCoevaluationMorphism
1.14-76 AddCoclosedMonoidalLeftCoevaluationMorphismWithGivenSource
1.14-77 AddCoclosedMonoidalLeftEvaluationMorphism
1.14-78 AddCoclosedMonoidalLeftEvaluationMorphismWithGivenRange
1.14-79 AddCoclosedMonoidalRightCoevaluationMorphism
1.14-80 AddCoclosedMonoidalRightCoevaluationMorphismWithGivenSource
1.14-81 AddCoclosedMonoidalRightEvaluationMorphism
1.14-82 AddCoclosedMonoidalRightEvaluationMorphismWithGivenRange
1.14-83 AddInternalCoHomOnMorphisms
1.14-84 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms
1.14-85 AddInternalCoHomOnObjects
1.14-86 AddInternalCoHomTensorProductCompatibilityMorphism
1.14-87 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
1.14-88 AddInternalCoHomToTensorProductLeftAdjunctMorphism
1.14-89 AddInternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct
1.14-90 AddInternalCoHomToTensorProductRightAdjunctMorphism
1.14-91 AddInternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct
1.14-92 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
1.14-93 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
1.14-94 AddIsomorphismFromInternalCoHomToObject
1.14-95 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
1.14-96 AddIsomorphismFromObjectToInternalCoHom
1.14-97 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
1.14-98 AddMonoidalPostCoComposeMorphism
1.14-99 AddMonoidalPostCoComposeMorphismWithGivenObjects
1.14-100 AddMonoidalPreCoComposeMorphism
1.14-101 AddMonoidalPreCoComposeMorphismWithGivenObjects
1.14-102 AddMorphismFromCoBidual
1.14-103 AddMorphismFromCoBidualWithGivenCoBidual
1.14-104 AddMorphismFromInternalCoHomToTensorProduct
1.14-105 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects
1.14-106 AddTensorProductToInternalCoHomLeftAdjunctMorphism
1.14-107 AddTensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom
1.14-108 AddTensorProductToInternalCoHomRightAdjunctMorphism
1.14-109 AddTensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom
1.14-110 AddUniversalPropertyOfCoDual
1.14-111 AddIsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit
1.14-112 AddIsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject
1.14-113 AddIsomorphismFromLeftInternalHomToObject
1.14-114 AddIsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom
1.14-115 AddIsomorphismFromObjectToLeftInternalHom
1.14-116 AddIsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom
1.14-117 AddLeftClosedMonoidalCoevaluationMorphism
1.14-118 AddLeftClosedMonoidalCoevaluationMorphismWithGivenRange
1.14-119 AddLeftClosedMonoidalEvaluationForLeftDual
1.14-120 AddLeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct
1.14-121 AddLeftClosedMonoidalEvaluationMorphism
1.14-122 AddLeftClosedMonoidalEvaluationMorphismWithGivenSource
1.14-123 AddLeftClosedMonoidalLambdaElimination
1.14-124 AddLeftClosedMonoidalLambdaIntroduction
1.14-125 AddLeftClosedMonoidalPostComposeMorphism
1.14-126 AddLeftClosedMonoidalPostComposeMorphismWithGivenObjects
1.14-127 AddLeftClosedMonoidalPreComposeMorphism
1.14-128 AddLeftClosedMonoidalPreComposeMorphismWithGivenObjects
1.14-129 AddLeftDualOnMorphisms
1.14-130 AddLeftDualOnMorphismsWithGivenLeftDuals
1.14-131 AddLeftDualOnObjects
1.14-132 AddLeftInternalHomOnMorphisms
1.14-133 AddLeftInternalHomOnMorphismsWithGivenLeftInternalHoms
1.14-134 AddLeftInternalHomOnObjects
1.14-135 AddLeftInternalHomToTensorProductAdjunctMorphism
1.14-136 AddLeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct
1.14-137 AddMorphismFromTensorProductToLeftInternalHom
1.14-138 AddMorphismFromTensorProductToLeftInternalHomWithGivenObjects
1.14-139 AddMorphismToLeftBidual
1.14-140 AddMorphismToLeftBidualWithGivenLeftBidual
1.14-141 AddTensorProductLeftDualityCompatibilityMorphism
1.14-142 AddTensorProductLeftDualityCompatibilityMorphismWithGivenObjects
1.14-143 AddTensorProductLeftInternalHomCompatibilityMorphism
1.14-144 AddTensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects
1.14-145 AddTensorProductToLeftInternalHomAdjunctMorphism
1.14-146 AddTensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom
1.14-147 AddUniversalPropertyOfLeftDual
1.14-148 AddIsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit
1.14-149 AddIsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject
1.14-150 AddIsomorphismFromLeftInternalCoHomToObject
1.14-151 AddIsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom
1.14-152 AddIsomorphismFromObjectToLeftInternalCoHom
1.14-153 AddIsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom
1.14-154 AddLeftCoDualOnMorphisms
1.14-155 AddLeftCoDualOnMorphismsWithGivenLeftCoDuals
1.14-156 AddLeftCoDualOnObjects
1.14-157 AddLeftCoDualityTensorProductCompatibilityMorphism
1.14-158 AddLeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects
1.14-159 AddLeftCoclosedMonoidalCoevaluationMorphism
1.14-160 AddLeftCoclosedMonoidalCoevaluationMorphismWithGivenSource
1.14-161 AddLeftCoclosedMonoidalEvaluationForLeftCoDual
1.14-162 AddLeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct
1.14-163 AddLeftCoclosedMonoidalEvaluationMorphism
1.14-164 AddLeftCoclosedMonoidalEvaluationMorphismWithGivenRange
1.14-165 AddLeftCoclosedMonoidalLambdaElimination
1.14-166 AddLeftCoclosedMonoidalLambdaIntroduction
1.14-167 AddLeftCoclosedMonoidalPostCoComposeMorphism
1.14-168 AddLeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects
1.14-169 AddLeftCoclosedMonoidalPreCoComposeMorphism
1.14-170 AddLeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects
1.14-171 AddLeftInternalCoHomOnMorphisms
1.14-172 AddLeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms
1.14-173 AddLeftInternalCoHomOnObjects
1.14-174 AddLeftInternalCoHomTensorProductCompatibilityMorphism
1.14-175 AddLeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
1.14-176 AddLeftInternalCoHomToTensorProductAdjunctMorphism
1.14-177 AddLeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct
1.14-178 AddMorphismFromLeftCoBidual
1.14-179 AddMorphismFromLeftCoBidualWithGivenLeftCoBidual
1.14-180 AddMorphismFromLeftInternalCoHomToTensorProduct
1.14-181 AddMorphismFromLeftInternalCoHomToTensorProductWithGivenObjects
1.14-182 AddTensorProductToLeftInternalCoHomAdjunctMorphism
1.14-183 AddTensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom
1.14-184 AddUniversalPropertyOfLeftCoDual
1.14-185 AddAssociatorLeftToRight
1.14-186 AddAssociatorLeftToRightWithGivenTensorProducts
1.14-187 AddAssociatorRightToLeft
1.14-188 AddAssociatorRightToLeftWithGivenTensorProducts
1.14-189 AddLeftUnitor
1.14-190 AddLeftUnitorInverse
1.14-191 AddLeftUnitorInverseWithGivenTensorProduct
1.14-192 AddLeftUnitorWithGivenTensorProduct
1.14-193 AddRightUnitor
1.14-194 AddRightUnitorInverse
1.14-195 AddRightUnitorInverseWithGivenTensorProduct
1.14-196 AddRightUnitorWithGivenTensorProduct
1.14-197 AddTensorProductOnMorphisms
1.14-198 AddTensorProductOnMorphismsWithGivenTensorProducts
1.14-199 AddCoevaluationForDual
1.14-200 AddCoevaluationForDualWithGivenTensorProduct
1.14-201 AddIsomorphismFromInternalHomToTensorProductWithDualObject
1.14-202 AddIsomorphismFromTensorProductWithDualObjectToInternalHom
1.14-203 AddMorphismFromBidual
1.14-204 AddMorphismFromBidualWithGivenBidual
1.14-205 AddMorphismFromInternalHomToTensorProduct
1.14-206 AddMorphismFromInternalHomToTensorProductWithGivenObjects
1.14-207 AddRankMorphism
1.14-208 AddTensorProductInternalHomCompatibilityMorphismInverse
1.14-209 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
1.14-210 AddTraceMap
1.14-211 AddCoRankMorphism
1.14-212 AddCoTraceMap
1.14-213 AddCoclosedCoevaluationForCoDual
1.14-214 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct
1.14-215 AddInternalCoHomTensorProductCompatibilityMorphismInverse
1.14-216 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
1.14-217 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject
1.14-218 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
1.14-219 AddMorphismFromTensorProductToInternalCoHom
1.14-220 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects
1.14-221 AddMorphismToCoBidual
1.14-222 AddMorphismToCoBidualWithGivenCoBidual

### 1 Monoidal Categories

#### 1.1 Monoidal Categories

A 6-tuple ( \mathbf{C}, \otimes, 1, \alpha, \lambda, \rho ) consisting of

- a category \mathbf{C},

- a functor \otimes: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C} compatible with the congruence of morphisms,

- an object 1 \in \mathbf{C},

- a natural isomorphism \alpha_{a,b,c}: a \otimes (b \otimes c) \cong (a \otimes b) \otimes c,

- a natural isomorphism \lambda_{a}: 1 \otimes a \cong a,

- a natural isomorphism \rho_{a}: a \otimes 1 \cong a,

is called a monoidal category, if

- for all objects a,b,c,d, the pentagon identity holds:

(\alpha_{a,b,c} \otimes \mathrm{id}_d) \circ \alpha_{a,b \otimes c, d} \circ ( \mathrm{id}_a \otimes \alpha_{b,c,d} ) \sim \alpha_{a \otimes b, c, d} \circ \alpha_{a,b,c \otimes d},

- for all objects a,c, the triangle identity holds:

( \rho_a \otimes \mathrm{id}_c ) \circ \alpha_{a,1,c} \sim \mathrm{id}_a \otimes \lambda_c.

The corresponding GAP property is given by `IsMonoidalCategory`.

##### 1.1-1 TensorProductOnMorphisms

`‣ TensorProductOnMorphisms`( alpha, beta ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, a' \otimes b')

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the tensor product \alpha \otimes \beta.

##### 1.1-2 TensorProductOnMorphismsWithGivenTensorProducts

`‣ TensorProductOnMorphismsWithGivenTensorProducts`( s, alpha, beta, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, a' \otimes b')

The arguments are an object s = a \otimes b, two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = a' \otimes b'. The output is the tensor product \alpha \otimes \beta.

##### 1.1-3 AssociatorRightToLeft

`‣ AssociatorRightToLeft`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c ).

The arguments are three objects a,b,c. The output is the associator \alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c.

##### 1.1-4 AssociatorRightToLeftWithGivenTensorProducts

`‣ AssociatorRightToLeftWithGivenTensorProducts`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c ).

The arguments are an object s = a \otimes (b \otimes c), three objects a,b,c, and an object r = (a \otimes b) \otimes c. The output is the associator \alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c.

##### 1.1-5 AssociatorLeftToRight

`‣ AssociatorLeftToRight`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) ).

The arguments are three objects a,b,c. The output is the associator \alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c).

##### 1.1-6 AssociatorLeftToRightWithGivenTensorProducts

`‣ AssociatorLeftToRightWithGivenTensorProducts`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) ).

The arguments are an object s = (a \otimes b) \otimes c, three objects a,b,c, and an object r = a \otimes (b \otimes c). The output is the associator \alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c).

##### 1.1-7 LeftUnitor

`‣ LeftUnitor`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1 \otimes a, a)

The argument is an object a. The output is the left unitor \lambda_a: 1 \otimes a \rightarrow a.

##### 1.1-8 LeftUnitorWithGivenTensorProduct

`‣ LeftUnitorWithGivenTensorProduct`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(1 \otimes a, a)

The arguments are an object a and an object s = 1 \otimes a. The output is the left unitor \lambda_a: 1 \otimes a \rightarrow a.

##### 1.1-9 LeftUnitorInverse

`‣ LeftUnitorInverse`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, 1 \otimes a)

The argument is an object a. The output is the inverse of the left unitor \lambda_a^{-1}: a \rightarrow 1 \otimes a.

##### 1.1-10 LeftUnitorInverseWithGivenTensorProduct

`‣ LeftUnitorInverseWithGivenTensorProduct`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, 1 \otimes a)

The argument is an object a and an object r = 1 \otimes a. The output is the inverse of the left unitor \lambda_a^{-1}: a \rightarrow 1 \otimes a.

##### 1.1-11 RightUnitor

`‣ RightUnitor`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a \otimes 1, a)

The argument is an object a. The output is the right unitor \rho_a: a \otimes 1 \rightarrow a.

##### 1.1-12 RightUnitorWithGivenTensorProduct

`‣ RightUnitorWithGivenTensorProduct`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes 1, a)

The arguments are an object a and an object s = a \otimes 1. The output is the right unitor \rho_a: a \otimes 1 \rightarrow a.

##### 1.1-13 RightUnitorInverse

`‣ RightUnitorInverse`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, a \otimes 1)

The argument is an object a. The output is the inverse of the right unitor \rho_a^{-1}: a \rightarrow a \otimes 1.

##### 1.1-14 RightUnitorInverseWithGivenTensorProduct

`‣ RightUnitorInverseWithGivenTensorProduct`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, a \otimes 1)

The arguments are an object a and an object r = a \otimes 1. The output is the inverse of the right unitor \rho_a^{-1}: a \rightarrow a \otimes 1.

##### 1.1-15 TensorProductOnObjects

`‣ TensorProductOnObjects`( a, b ) | ( operation )

Returns: an object

The arguments are two objects a, b. The output is the tensor product a \otimes b.

##### 1.1-16 AddTensorProductOnObjects

`‣ AddTensorProductOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductOnObjects`. F: (a,b) \mapsto a \otimes b.

##### 1.1-17 TensorUnit

`‣ TensorUnit`( C ) | ( attribute )

Returns: an object

The argument is a category \mathbf{C}. The output is the tensor unit 1 of \mathbf{C}.

##### 1.1-18 AddTensorUnit

`‣ AddTensorUnit`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorUnit`. F: ( ) \mapsto 1.

#### 1.2 Additive Monoidal Categories

##### 1.2-1 LeftDistributivityExpanding

`‣ LeftDistributivityExpanding`( a, L ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes (b_1 \oplus \dots \oplus b_n), (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) )

The arguments are an object a and a list of objects L = (b_1, \dots, b_n). The output is the left distributivity morphism a \otimes (b_1 \oplus \dots \oplus b_n) \rightarrow (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n).

##### 1.2-2 LeftDistributivityExpandingWithGivenObjects

`‣ LeftDistributivityExpandingWithGivenObjects`( s, a, L, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = a \otimes (b_1 \oplus \dots \oplus b_n), an object a, a list of objects L = (b_1, \dots, b_n), and an object r = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n). The output is the left distributivity morphism s \rightarrow r.

##### 1.2-3 LeftDistributivityFactoring

`‣ LeftDistributivityFactoring`( a, L ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), a \otimes (b_1 \oplus \dots \oplus b_n) )

The arguments are an object a and a list of objects L = (b_1, \dots, b_n). The output is the left distributivity morphism (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) \rightarrow a \otimes (b_1 \oplus \dots \oplus b_n).

##### 1.2-4 LeftDistributivityFactoringWithGivenObjects

`‣ LeftDistributivityFactoringWithGivenObjects`( s, a, L, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), an object a, a list of objects L = (b_1, \dots, b_n), and an object r = a \otimes (b_1 \oplus \dots \oplus b_n). The output is the left distributivity morphism s \rightarrow r.

##### 1.2-5 RightDistributivityExpanding

`‣ RightDistributivityExpanding`( L, a ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (b_1 \oplus \dots \oplus b_n) \otimes a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) )

The arguments are a list of objects L = (b_1, \dots, b_n) and an object a. The output is the right distributivity morphism (b_1 \oplus \dots \oplus b_n) \otimes a \rightarrow (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a).

##### 1.2-6 RightDistributivityExpandingWithGivenObjects

`‣ RightDistributivityExpandingWithGivenObjects`( s, L, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (b_1 \oplus \dots \oplus b_n) \otimes a, a list of objects L = (b_1, \dots, b_n), an object a, and an object r = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a). The output is the right distributivity morphism s \rightarrow r.

##### 1.2-7 RightDistributivityFactoring

`‣ RightDistributivityFactoring`( L, a ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), (b_1 \oplus \dots \oplus b_n) \otimes a)

The arguments are a list of objects L = (b_1, \dots, b_n) and an object a. The output is the right distributivity morphism (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) \rightarrow (b_1 \oplus \dots \oplus b_n) \otimes a .

##### 1.2-8 RightDistributivityFactoringWithGivenObjects

`‣ RightDistributivityFactoringWithGivenObjects`( s, L, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), a list of objects L = (b_1, \dots, b_n), an object a, and an object r = (b_1 \oplus \dots \oplus b_n) \otimes a. The output is the right distributivity morphism s \rightarrow r.

#### 1.3 Braided Monoidal Categories

A monoidal category \mathbf{C} equipped with a natural isomorphism B_{a,b}: a \otimes b \cong b \otimes a is called a braided monoidal category if

- \lambda_a \circ B_{a,1} \sim \rho_a,

- (B_{c,a} \otimes \mathrm{id}_b) \circ \alpha_{c,a,b} \circ B_{a \otimes b,c} \sim \alpha_{a,c,b} \circ ( \mathrm{id}_a \otimes B_{b,c}) \circ \alpha^{-1}_{a,b,c},

- ( \mathrm{id}_b \otimes B_{c,a} ) \circ \alpha^{-1}_{b,c,a} \circ B_{a,b \otimes c} \sim \alpha^{-1}_{b,a,c} \circ (B_{a,b} \otimes \mathrm{id}_c) \circ \alpha_{a,b,c}.

The corresponding GAP property is given by `IsBraidedMonoidalCategory`.

##### 1.3-1 Braiding

`‣ Braiding`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes b, b \otimes a ).

The arguments are two objects a,b. The output is the braiding B_{a,b}: a \otimes b \rightarrow b \otimes a.

##### 1.3-2 BraidingWithGivenTensorProducts

`‣ BraidingWithGivenTensorProducts`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes b, b \otimes a ).

The arguments are an object s = a \otimes b, two objects a,b, and an object r = b \otimes a. The output is the braiding B_{a,b}: a \otimes b \rightarrow b \otimes a.

##### 1.3-3 BraidingInverse

`‣ BraidingInverse`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b \otimes a, a \otimes b ).

The arguments are two objects a,b. The output is the inverse braiding B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b.

##### 1.3-4 BraidingInverseWithGivenTensorProducts

`‣ BraidingInverseWithGivenTensorProducts`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b \otimes a, a \otimes b ).

The arguments are an object s = b \otimes a, two objects a,b, and an object r = a \otimes b. The output is the inverse braiding B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b.

#### 1.4 Symmetric Monoidal Categories

A braided monoidal category \mathbf{C} is called symmetric monoidal category if B_{a,b}^{-1} \sim B_{b,a}. The corresponding GAP property is given by `IsSymmetricMonoidalCategory`.

#### 1.5 Left Closed Monoidal Categories

A monoidal category \mathbf{C} which has for each functor - \otimes b: \mathbf{C} \rightarrow \mathbf{C} a right adjoint (denoted by \mathrm{\underline{Hom}_\ell}(b,-)) is called a left closed monoidal category.

If no operations involving left duals are installed manually, the left dual objects will be derived as a^\vee \coloneqq \mathrm{\underline{Hom}_\ell}(a,1).

The corresponding GAP property is called `IsLeftClosedMonoidalCategory`.

##### 1.5-1 LeftInternalHomOnObjects

`‣ LeftInternalHomOnObjects`( a, b ) | ( operation )

Returns: an object

The arguments are two objects a,b. The output is the internal hom object \mathrm{\underline{Hom}_\ell}(a,b).

##### 1.5-2 LeftInternalHomOnMorphisms

`‣ LeftInternalHomOnMorphisms`( alpha, beta ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a',b), \mathrm{\underline{Hom}_\ell}(a,b') )

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the internal hom morphism \mathrm{\underline{Hom}_\ell}(\alpha,\beta): \mathrm{\underline{Hom}_\ell}(a',b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,b').

##### 1.5-3 LeftInternalHomOnMorphismsWithGivenLeftInternalHoms

`‣ LeftInternalHomOnMorphismsWithGivenLeftInternalHoms`( s, alpha, beta, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = \mathrm{\underline{Hom}_\ell}(a',b), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{Hom}_\ell}(a,b'). The output is the internal hom morphism \mathrm{\underline{Hom}_\ell}(\alpha,\beta): \mathrm{\underline{Hom}_\ell}(a',b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,b').

##### 1.5-4 LeftClosedMonoidalEvaluationMorphism

`‣ LeftClosedMonoidalEvaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a,b) \otimes a, b ).

The arguments are two objects a, b. The output is the evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.

##### 1.5-5 LeftClosedMonoidalEvaluationMorphismWithGivenSource

`‣ LeftClosedMonoidalEvaluationMorphismWithGivenSource`( a, b, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, b ).

The arguments are two objects a,b and an object s = \mathrm{\underline{Hom}_\ell}(a,b) \otimes a. The output is the evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.

##### 1.5-6 LeftClosedMonoidalCoevaluationMorphism

`‣ LeftClosedMonoidalCoevaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{Hom}_\ell}(a, b \otimes a) ).

The arguments are two objects a,b. The output is the coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}_\ell}(a, b \otimes a), i.e., the unit of the tensor hom adjunction.

##### 1.5-7 LeftClosedMonoidalCoevaluationMorphismWithGivenRange

`‣ LeftClosedMonoidalCoevaluationMorphismWithGivenRange`( a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, r ).

The arguments are two objects a,b and an object r = \mathrm{\underline{Hom}_\ell}(a, b \otimes a). The output is the coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}_\ell}(a, b \otimes a), i.e., the unit of the tensor hom adjunction.

##### 1.5-8 TensorProductToLeftInternalHomAdjunctMorphism

`‣ TensorProductToLeftInternalHomAdjunctMorphism`( a, b, f ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}_\ell}(b,c) ).

The arguments are two objects a,b and a morphism f: a \otimes b \rightarrow c. The output is a morphism g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c) corresponding to f under the tensor hom adjunction.

##### 1.5-9 TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom

`‣ TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom`( a, b, f, i ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, i ).

The arguments are two objects a,b, a morphism f: a \otimes b \rightarrow c and an object i = \mathrm{\underline{Hom}_\ell}(b,c). The output is a morphism g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c) corresponding to f under the tensor hom adjunction.

##### 1.5-10 LeftInternalHomToTensorProductAdjunctMorphism

`‣ LeftInternalHomToTensorProductAdjunctMorphism`( b, c, g ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, c).

The arguments are two objects b,c and a morphism g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c). The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.

##### 1.5-11 LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct

`‣ LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct`( b, c, g, t ) | ( operation )

Returns: a morphism in \mathrm{Hom}(t, c).

The arguments are two objects b,c, a morphism g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c) and an object t = a \otimes b. The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.

##### 1.5-12 LeftClosedMonoidalPreComposeMorphism

`‣ LeftClosedMonoidalPreComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c), \mathrm{\underline{Hom}_\ell}(a,c) ).

The arguments are three objects a,b,c. The output is the precomposition morphism \mathrm{LeftClosedMonoidalPreComposeMorphism}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c).

##### 1.5-13 LeftClosedMonoidalPreComposeMorphismWithGivenObjects

`‣ LeftClosedMonoidalPreComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c), three objects a,b,c, and an object r = \mathrm{\underline{Hom}_\ell}(a,c). The output is the precomposition morphism \mathrm{LeftClosedMonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c).

##### 1.5-14 LeftClosedMonoidalPostComposeMorphism

`‣ LeftClosedMonoidalPostComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b), \mathrm{\underline{Hom}_\ell}(a,c) ).

The arguments are three objects a,b,c. The output is the postcomposition morphism \mathrm{LeftClosedMonoidalPostComposeMorphism}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c).

##### 1.5-15 LeftClosedMonoidalPostComposeMorphismWithGivenObjects

`‣ LeftClosedMonoidalPostComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b), three objects a,b,c, and an object r = \mathrm{\underline{Hom}_\ell}(a,c). The output is the postcomposition morphism \mathrm{LeftClosedMonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c).

##### 1.5-16 LeftDualOnObjects

`‣ LeftDualOnObjects`( a ) | ( attribute )

Returns: an object

The argument is an object a. The output is its dual object a^{\vee}.

##### 1.5-17 LeftDualOnMorphisms

`‣ LeftDualOnMorphisms`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( b^{\vee}, a^{\vee} ).

The argument is a morphism \alpha: a \rightarrow b. The output is its dual morphism \alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.5-18 LeftDualOnMorphismsWithGivenLeftDuals

`‣ LeftDualOnMorphismsWithGivenLeftDuals`( s, alpha, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The argument is an object s = b^{\vee}, a morphism \alpha: a \rightarrow b, and an object r = a^{\vee}. The output is the dual morphism \alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.5-19 LeftClosedMonoidalEvaluationForLeftDual

`‣ LeftClosedMonoidalEvaluationForLeftDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes a, 1 ).

The argument is an object a. The output is the evaluation morphism \mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1.

##### 1.5-20 LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct

`‣ LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \otimes a, an object a, and an object r = 1. The output is the evaluation morphism \mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1.

##### 1.5-21 MorphismToLeftBidual

`‣ MorphismToLeftBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, (a^{\vee})^{\vee}).

The argument is an object a. The output is the morphism to the bidual a \rightarrow (a^{\vee})^{\vee}.

##### 1.5-22 MorphismToLeftBidualWithGivenLeftBidual

`‣ MorphismToLeftBidualWithGivenLeftBidual`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, r).

The arguments are an object a, and an object r = (a^{\vee})^{\vee}. The output is the morphism to the bidual a \rightarrow (a^{\vee})^{\vee}.

##### 1.5-23 TensorProductLeftInternalHomCompatibilityMorphism

`‣ TensorProductLeftInternalHomCompatibilityMorphism`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b'), \mathrm{\underline{Hom}_\ell}(a \otimes b,a' \otimes b')).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{TensorProductLeftInternalHomCompatibilityMorphism}_{a,a',b,b'}: \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b') \rightarrow \mathrm{\underline{Hom}_\ell}(a \otimes b,a' \otimes b').

##### 1.5-24 TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects

`‣ TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b') and r = \mathrm{\underline{Hom}_\ell}(a \otimes b,a' \otimes b'). The output is the natural morphism \mathrm{TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b') \rightarrow \mathrm{\underline{Hom}_\ell}(a \otimes b,a' \otimes b').

##### 1.5-25 TensorProductLeftDualityCompatibilityMorphism

`‣ TensorProductLeftDualityCompatibilityMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{TensorProductLeftDualityCompatibilityMorphism}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.

##### 1.5-26 TensorProductLeftDualityCompatibilityMorphismWithGivenObjects

`‣ TensorProductLeftDualityCompatibilityMorphismWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \otimes b^{\vee}, two objects a,b, and an object r = (a \otimes b)^{\vee}. The output is the natural morphism \mathrm{TensorProductLeftDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.

##### 1.5-27 MorphismFromTensorProductToLeftInternalHom

`‣ MorphismFromTensorProductToLeftInternalHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}_\ell}(a,b) ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromTensorProductToLeftInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}_\ell}(a,b).

##### 1.5-28 MorphismFromTensorProductToLeftInternalHomWithGivenObjects

`‣ MorphismFromTensorProductToLeftInternalHomWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \otimes b, two objects a,b, and an object r = \mathrm{\underline{Hom}_\ell}(a,b). The output is the natural morphism \mathrm{MorphismFromTensorProductToLeftInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}_\ell}(a,b).

##### 1.5-29 IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit

`‣ IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a^{\vee}, \mathrm{\underline{Hom}_\ell}(a,1)).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit}_{a}: a^{\vee} \rightarrow \mathrm{\underline{Hom}_\ell}(a,1).

##### 1.5-30 IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject

`‣ IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}_\ell}(a,1), a^{\vee}).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject}_{a}: \mathrm{\underline{Hom}_\ell}(a,1) \rightarrow a^{\vee}.

##### 1.5-31 UniversalPropertyOfLeftDual

`‣ UniversalPropertyOfLeftDual`( t, a, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(t, a^{\vee}).

The arguments are two objects t,a, and a morphism \alpha: t \otimes a \rightarrow 1. The output is the morphism t \rightarrow a^{\vee} given by the universal property of a^{\vee}.

##### 1.5-32 LeftClosedMonoidalLambdaIntroduction

`‣ LeftClosedMonoidalLambdaIntroduction`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( 1, \mathrm{\underline{Hom}_\ell}(a,b) ).

The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism 1 \rightarrow \mathrm{\underline{Hom}_\ell}(a,b) under the tensor hom adjunction.

##### 1.5-33 LeftClosedMonoidalLambdaElimination

`‣ LeftClosedMonoidalLambdaElimination`( a, b, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a,b).

The arguments are two objects a,b, and a morphism \alpha: 1 \rightarrow \mathrm{\underline{Hom}_\ell}(a,b). The output is a morphism a \rightarrow b corresponding to \alpha under the tensor hom adjunction.

##### 1.5-34 IsomorphismFromObjectToLeftInternalHom

`‣ IsomorphismFromObjectToLeftInternalHom`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{Hom}_\ell}(1,a)).

The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{Hom}_\ell}(1,a).

##### 1.5-35 IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom

`‣ IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, r).

The argument is an object a, and an object r = \mathrm{\underline{Hom}_\ell}(1,a). The output is the natural isomorphism a \rightarrow \mathrm{\underline{Hom}_\ell}(1,a).

##### 1.5-36 IsomorphismFromLeftInternalHomToObject

`‣ IsomorphismFromLeftInternalHomToObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}_\ell}(1,a),a).

The argument is an object a. The output is the natural isomorphism \mathrm{\underline{Hom}_\ell}(1,a) \rightarrow a.

##### 1.5-37 IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom

`‣ IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(s,a).

The argument is an object a, and an object s = \mathrm{\underline{Hom}_\ell}(1,a). The output is the natural isomorphism \mathrm{\underline{Hom}_\ell}(1,a) \rightarrow a.

#### 1.6 Closed Monoidal Categories

A monoidal category \mathbf{C} which has for each functor - \otimes b: \mathbf{C} \rightarrow \mathbf{C} a right adjoint (denoted by \mathrm{\underline{Hom}_\ell}(b,-)) is called a closed monoidal category.

If no operations involving duals are installed manually, the dual objects will be derived as a^\vee \coloneqq \mathrm{\underline{Hom}_\ell}(a,1).

The corresponding GAP property is called `IsClosedMonoidalCategory`.

##### 1.6-1 InternalHomOnObjects

`‣ InternalHomOnObjects`( a, b ) | ( operation )

Returns: an object

The arguments are two objects a,b. The output is the internal hom object \mathrm{\underline{Hom}}(a,b).

##### 1.6-2 InternalHomOnMorphisms

`‣ InternalHomOnMorphisms`( alpha, beta ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a',b), \mathrm{\underline{Hom}}(a,b') )

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the internal hom morphism \mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b').

##### 1.6-3 InternalHomOnMorphismsWithGivenInternalHoms

`‣ InternalHomOnMorphismsWithGivenInternalHoms`( s, alpha, beta, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = \mathrm{\underline{Hom}}(a',b), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{Hom}}(a,b'). The output is the internal hom morphism \mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b').

##### 1.6-4 ClosedMonoidalRightEvaluationMorphism

`‣ ClosedMonoidalRightEvaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes \mathrm{\underline{Hom}}(a,b), b ).

The arguments are two objects a, b. The output is the right evaluation morphism \mathrm{ev}_{a,b}:a \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow b, i.e., the counit of the tensor hom adjunction.

##### 1.6-5 ClosedMonoidalRightEvaluationMorphismWithGivenSource

`‣ ClosedMonoidalRightEvaluationMorphismWithGivenSource`( a, b, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, b ).

The arguments are two objects a,b and an object s = a \otimes \mathrm{\underline{Hom}}(a,b). The output is the right evaluation morphism \mathrm{ev}_{a,b}: a \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow b, i.e., the counit of the tensor hom adjunction.

##### 1.6-6 ClosedMonoidalRightCoevaluationMorphism

`‣ ClosedMonoidalRightCoevaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{Hom}}(a, a \otimes b) ).

The arguments are two objects a,b. The output is the right coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}}(a, a \otimes b), i.e., the unit of the tensor hom adjunction.

##### 1.6-7 ClosedMonoidalRightCoevaluationMorphismWithGivenRange

`‣ ClosedMonoidalRightCoevaluationMorphismWithGivenRange`( a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, r ).

The arguments are two objects a,b and an object r = \mathrm{\underline{Hom}}(a, a \otimes b). The output is the right coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}}(a, a \otimes b), i.e., the unit of the tensor hom adjunction.

##### 1.6-8 TensorProductToInternalHomRightAdjunctMorphism

`‣ TensorProductToInternalHomRightAdjunctMorphism`( a, b, f ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{Hom}}(a,c) ).

The arguments are two objects a,b and a morphism f: a \otimes b \rightarrow c. The output is a morphism g: b \rightarrow \mathrm{\underline{Hom}}(a,c) corresponding to f under the tensor hom adjunction.

##### 1.6-9 TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom

`‣ TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom`( a, b, f, i ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, i ).

The arguments are two objects a,b, a morphism f: a \otimes b \rightarrow c and an object i = \mathrm{\underline{Hom}}(a,c). The output is a morphism g: b \rightarrow i corresponding to f under the tensor hom adjunction.

##### 1.6-10 TensorProductToInternalHomRightAdjunctionIsomorphism

`‣ TensorProductToInternalHomRightAdjunctionIsomorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( H(a \otimes b, c), H(b, \mathrm{\underline{Hom}}(a,c)) ).

The arguments are three objects a,b,c. The output is the tri-natural isomorphism H(a \otimes b, c) \to H(b, \mathrm{\underline{Hom}}(a,c)) in the range category of the homomorphism structure H.

##### 1.6-11 TensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects

`‣ TensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are fives objects s,a,b,c,r where s = H(a \otimes b, c) and r = H(b, \mathrm{\underline{Hom}}(a,c)). The output is the tri-natural isomorphism s \to r in the range category of the homomorphism structure H.

##### 1.6-12 InternalHomToTensorProductRightAdjunctMorphism

`‣ InternalHomToTensorProductRightAdjunctMorphism`( a, c, g ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, c).

The arguments are two objects a,c and a morphism g: b \rightarrow \mathrm{\underline{Hom}}(a,c). The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.

##### 1.6-13 InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct

`‣ InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct`( a, c, g, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(s, c).

The arguments are two objects a,c, a morphism g: b \rightarrow \mathrm{\underline{Hom}}(a,c) and an object s = a \otimes b. The output is a morphism f: s \rightarrow c corresponding to g under the tensor hom adjunction.

##### 1.6-14 InternalHomToTensorProductRightAdjunctionIsomorphism

`‣ InternalHomToTensorProductRightAdjunctionIsomorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( H(b, \mathrm{\underline{Hom}}(a,c)), H(a \otimes b, c) ).

The arguments are three objects a,b,c. The output is the tri-natural isomorphism H(b, \mathrm{\underline{Hom}}(a,c)) \to H(a \otimes b, c) in the range category of the homomorphism structure H.

##### 1.6-15 InternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects

`‣ InternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are fives objects s,a,b,c,r where s = H(b, \mathrm{\underline{Hom}}(a,c)) and r = H(a \otimes b, c). The output is the tri-natural isomorphism s \to r in the range category of the homomorphism structure H.

##### 1.6-16 ClosedMonoidalLeftEvaluationMorphism

`‣ ClosedMonoidalLeftEvaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b ).

The arguments are two objects a, b. The output is the left evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.

##### 1.6-17 ClosedMonoidalLeftEvaluationMorphismWithGivenSource

`‣ ClosedMonoidalLeftEvaluationMorphismWithGivenSource`( a, b, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, b ).

The arguments are two objects a,b and an object s = \mathrm{\underline{Hom}}(a,b) \otimes a. The output is the left evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.

##### 1.6-18 ClosedMonoidalLeftCoevaluationMorphism

`‣ ClosedMonoidalLeftCoevaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{Hom}}(a, b \otimes a) ).

The arguments are two objects a,b. The output is the left coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}}(a, b \otimes a), i.e., the unit of the tensor hom adjunction.

##### 1.6-19 ClosedMonoidalLeftCoevaluationMorphismWithGivenRange

`‣ ClosedMonoidalLeftCoevaluationMorphismWithGivenRange`( a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, r ).

The arguments are two objects a,b and an object r = \mathrm{\underline{Hom}}(a, b \otimes a). The output is the left coevaluation morphism \mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}}(a, b \otimes a), i.e., the unit of the tensor hom adjunction.

##### 1.6-20 TensorProductToInternalHomLeftAdjunctMorphism

`‣ TensorProductToInternalHomLeftAdjunctMorphism`( a, b, f ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) ).

The arguments are two objects a,b and a morphism f: a \otimes b \rightarrow c. The output is a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c) corresponding to f under the tensor hom adjunction.

##### 1.6-21 TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom

`‣ TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom`( a, b, f, i ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, i ).

The arguments are two objects a,b, a morphism f: a \otimes b \rightarrow c and an object i = \mathrm{\underline{Hom}}(b,c). The output is a morphism g: a \rightarrow i corresponding to f under the tensor hom adjunction.

##### 1.6-22 TensorProductToInternalHomLeftAdjunctionIsomorphism

`‣ TensorProductToInternalHomLeftAdjunctionIsomorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( H(a \otimes b, c), H(a, \mathrm{\underline{Hom}}(b,c)) ).

The arguments are three objects a,b,c. The output is the tri-natural isomorphism H(a \otimes b, c) \to H(a, \mathrm{\underline{Hom}}(b,c)) in the range category of the homomorphism structure H.

##### 1.6-23 TensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects

`‣ TensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are fives objects s,a,b,c,r where s = H(a \otimes b, c) and r = H(a, \mathrm{\underline{Hom}}(b,c)). The output is the tri-natural isomorphism s \to r in the range category of the homomorphism structure H.

##### 1.6-24 InternalHomToTensorProductLeftAdjunctMorphism

`‣ InternalHomToTensorProductLeftAdjunctMorphism`( b, c, g ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, c).

The arguments are two objects b,c and a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c). The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.

##### 1.6-25 InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct

`‣ InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct`( b, c, g, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(s, c).

The arguments are two objects b,c, a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c) and an object s = a \otimes b. The output is a morphism f: s \rightarrow c corresponding to g under the tensor hom adjunction.

##### 1.6-26 InternalHomToTensorProductLeftAdjunctionIsomorphism

`‣ InternalHomToTensorProductLeftAdjunctionIsomorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( H(a, \mathrm{\underline{Hom}}(b,c)), H(a \otimes b, c) ).

The arguments are three objects a,b,c. The output is the tri-natural isomorphism H(a, \mathrm{\underline{Hom}}(b,c)) \to H(a \otimes b, c) in the range category of the homomorphism structure H.

##### 1.6-27 InternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects

`‣ InternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are fives objects s,a,b,c,r where s = H(a, \mathrm{\underline{Hom}}(b,c)) and r = H(a \otimes b, c). The output is the tri-natural isomorphism s \to r in the range category of the homomorphism structure H.

##### 1.6-28 MonoidalPreComposeMorphism

`‣ MonoidalPreComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) ).

The arguments are three objects a,b,c. The output is the precomposition morphism \mathrm{MonoidalPreComposeMorphism}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c).

##### 1.6-29 MonoidalPreComposeMorphismWithGivenObjects

`‣ MonoidalPreComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), three objects a,b,c, and an object r = \mathrm{\underline{Hom}}(a,c). The output is the precomposition morphism \mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c).

##### 1.6-30 MonoidalPostComposeMorphism

`‣ MonoidalPostComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) ).

The arguments are three objects a,b,c. The output is the postcomposition morphism \mathrm{MonoidalPostComposeMorphism}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c).

##### 1.6-31 MonoidalPostComposeMorphismWithGivenObjects

`‣ MonoidalPostComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), three objects a,b,c, and an object r = \mathrm{\underline{Hom}}(a,c). The output is the postcomposition morphism \mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c).

##### 1.6-32 DualOnObjects

`‣ DualOnObjects`( a ) | ( attribute )

Returns: an object

The argument is an object a. The output is its dual object a^{\vee}.

##### 1.6-33 DualOnMorphisms

`‣ DualOnMorphisms`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( b^{\vee}, a^{\vee} ).

The argument is a morphism \alpha: a \rightarrow b. The output is its dual morphism \alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.6-34 DualOnMorphismsWithGivenDuals

`‣ DualOnMorphismsWithGivenDuals`( s, alpha, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The argument is an object s = b^{\vee}, a morphism \alpha: a \rightarrow b, and an object r = a^{\vee}. The output is the dual morphism \alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.6-35 EvaluationForDual

`‣ EvaluationForDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes a, 1 ).

The argument is an object a. The output is the evaluation morphism \mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1.

##### 1.6-36 EvaluationForDualWithGivenTensorProduct

`‣ EvaluationForDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \otimes a, an object a, and an object r = 1. The output is the evaluation morphism \mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1.

##### 1.6-37 MorphismToBidual

`‣ MorphismToBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, (a^{\vee})^{\vee}).

The argument is an object a. The output is the morphism to the bidual a \rightarrow (a^{\vee})^{\vee}.

##### 1.6-38 MorphismToBidualWithGivenBidual

`‣ MorphismToBidualWithGivenBidual`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, r).

The arguments are an object a, and an object r = (a^{\vee})^{\vee}. The output is the morphism to the bidual a \rightarrow (a^{\vee})^{\vee}.

##### 1.6-39 TensorProductInternalHomCompatibilityMorphism

`‣ TensorProductInternalHomCompatibilityMorphism`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{TensorProductInternalHomCompatibilityMorphism}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b').

##### 1.6-40 TensorProductInternalHomCompatibilityMorphismWithGivenObjects

`‣ TensorProductInternalHomCompatibilityMorphismWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') and r = \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'). The output is the natural morphism \mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b').

##### 1.6-41 TensorProductDualityCompatibilityMorphism

`‣ TensorProductDualityCompatibilityMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{TensorProductDualityCompatibilityMorphism}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.

##### 1.6-42 TensorProductDualityCompatibilityMorphismWithGivenObjects

`‣ TensorProductDualityCompatibilityMorphismWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \otimes b^{\vee}, two objects a,b, and an object r = (a \otimes b)^{\vee}. The output is the natural morphism \mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.

##### 1.6-43 MorphismFromTensorProductToInternalHom

`‣ MorphismFromTensorProductToInternalHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromTensorProductToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b).

##### 1.6-44 MorphismFromTensorProductToInternalHomWithGivenObjects

`‣ MorphismFromTensorProductToInternalHomWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = a^{\vee} \otimes b, two objects a,b, and an object r = \mathrm{\underline{Hom}}(a,b). The output is the natural morphism \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b).

##### 1.6-45 IsomorphismFromDualObjectToInternalHomIntoTensorUnit

`‣ IsomorphismFromDualObjectToInternalHomIntoTensorUnit`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a^{\vee}, \mathrm{\underline{Hom}}(a,1)).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}_{a}: a^{\vee} \rightarrow \mathrm{\underline{Hom}}(a,1).

##### 1.6-46 IsomorphismFromInternalHomIntoTensorUnitToDualObject

`‣ IsomorphismFromInternalHomIntoTensorUnitToDualObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}}(a,1), a^{\vee}).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromInternalHomIntoTensorUnitToDualObject}_{a}: \mathrm{\underline{Hom}}(a,1) \rightarrow a^{\vee}.

##### 1.6-47 UniversalPropertyOfDual

`‣ UniversalPropertyOfDual`( t, a, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(t, a^{\vee}).

The arguments are two objects t,a, and a morphism \alpha: t \otimes a \rightarrow 1. The output is the morphism t \rightarrow a^{\vee} given by the universal property of a^{\vee}.

##### 1.6-48 LambdaIntroduction

`‣ LambdaIntroduction`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( 1, \mathrm{\underline{Hom}}(a,b) ).

The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism 1 \rightarrow \mathrm{\underline{Hom}}(a,b) under the tensor hom adjunction.

##### 1.6-49 LambdaElimination

`‣ LambdaElimination`( a, b, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a,b).

The arguments are two objects a,b, and a morphism \alpha: 1 \rightarrow \mathrm{\underline{Hom}}(a,b). The output is a morphism a \rightarrow b corresponding to \alpha under the tensor hom adjunction.

##### 1.6-50 IsomorphismFromObjectToInternalHom

`‣ IsomorphismFromObjectToInternalHom`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a)).

The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{Hom}}(1,a).

##### 1.6-51 IsomorphismFromObjectToInternalHomWithGivenInternalHom

`‣ IsomorphismFromObjectToInternalHomWithGivenInternalHom`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, r).

The argument is an object a, and an object r = \mathrm{\underline{Hom}}(1,a). The output is the natural isomorphism a \rightarrow \mathrm{\underline{Hom}}(1,a).

##### 1.6-52 IsomorphismFromInternalHomToObject

`‣ IsomorphismFromInternalHomToObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a).

The argument is an object a. The output is the natural isomorphism \mathrm{\underline{Hom}}(1,a) \rightarrow a.

##### 1.6-53 IsomorphismFromInternalHomToObjectWithGivenInternalHom

`‣ IsomorphismFromInternalHomToObjectWithGivenInternalHom`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(s,a).

The argument is an object a, and an object s = \mathrm{\underline{Hom}}(1,a). The output is the natural isomorphism \mathrm{\underline{Hom}}(1,a) \rightarrow a.

#### 1.7 Left Coclosed Monoidal Categories

A monoidal category \mathbf{C} which has for each functor - \otimes b: \mathbf{C} \rightarrow \mathbf{C} a left adjoint (denoted by \mathrm{\underline{coHom}}(-,b)) is called a left coclosed monoidal category.

If no operations involving left coduals are installed manually, the left codual objects will be derived as a_\vee \coloneqq \mathrm{\underline{coHom}}(1,a).

The corresponding GAP property is called `IsLeftCoclosedMonoidalCategory`.

##### 1.7-1 LeftInternalCoHomOnObjects

`‣ LeftInternalCoHomOnObjects`( a, b ) | ( operation )

Returns: an object

The arguments are two objects a,b. The output is the internal cohom object \mathrm{\underline{coHom}_\ell}(a,b).

##### 1.7-2 LeftInternalCoHomOnMorphisms

`‣ LeftInternalCoHomOnMorphisms`( alpha, beta ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,b'), \mathrm{\underline{coHom}_\ell}(a',b) )

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the internal cohom morphism \mathrm{\underline{coHom}_\ell}(\alpha,\beta): \mathrm{\underline{coHom}_\ell}(a,b') \rightarrow \mathrm{\underline{coHom}_\ell}(a',b).

##### 1.7-3 LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms

`‣ LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms`( s, alpha, beta, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = \mathrm{\underline{coHom}_\ell}(a,b'), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{coHom}_\ell}(a',b). The output is the internal cohom morphism \mathrm{\underline{coHom}_\ell}(\alpha,\beta): \mathrm{\underline{coHom}_\ell}(a,b') \rightarrow \mathrm{\underline{coHom}_\ell}(a',b).

##### 1.7-4 LeftCoclosedMonoidalEvaluationMorphism

`‣ LeftCoclosedMonoidalEvaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{coHom}_\ell}(b,a) \otimes a ).

The arguments are two objects a, b. The output is the coclosed evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow \mathrm{\underline{coHom}_\ell}(b,a) \otimes a, i.e., the unit of the cohom tensor adjunction.

##### 1.7-5 LeftCoclosedMonoidalEvaluationMorphismWithGivenRange

`‣ LeftCoclosedMonoidalEvaluationMorphismWithGivenRange`( a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, r ).

The arguments are two objects a,b and an object r = \mathrm{\underline{coHom}_\ell}(b,a) \otimes a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow \mathrm{\underline{coHom}_\ell}(b,a) \otimes a, i.e., the unit of the cohom tensor adjunction.

##### 1.7-6 LeftCoclosedMonoidalCoevaluationMorphism

`‣ LeftCoclosedMonoidalCoevaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(b \otimes a, a), b ).

The arguments are two objects a,b. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}_\ell}(b \otimes a, a) \rightarrow b, i.e., the counit of the cohom tensor adjunction.

##### 1.7-7 LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource

`‣ LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource`( a, b, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, b ).

The arguments are two objects a,b and an object s = \mathrm{\underline{coHom}_\ell}(b \otimes a, a). The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}_\ell}(b \otimes a, a) \rightarrow b, i.e., the unit of the cohom tensor adjunction.

##### 1.7-8 TensorProductToLeftInternalCoHomAdjunctMorphism

`‣ TensorProductToLeftInternalCoHomAdjunctMorphism`( b, c, g ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,c), b ).

The arguments are two objects b,c and a morphism g: a \rightarrow b \otimes c. The output is a morphism f: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow b corresponding to g under the cohom tensor adjunction.

##### 1.7-9 TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom

`‣ TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom`( b, c, g, i ) | ( operation )

Returns: a morphism in \mathrm{Hom}( i, b ).

The arguments are two objects b,c, a morphism g: a \rightarrow b \otimes c and an object i = \mathrm{\underline{coHom}_\ell}(a,c). The output is a morphism f: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow b corresponding to g under the cohom tensor adjunction.

##### 1.7-10 LeftInternalCoHomToTensorProductAdjunctMorphism

`‣ LeftInternalCoHomToTensorProductAdjunctMorphism`( a, c, f ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, b \otimes c).

The arguments are two objects a,c and a morphism f: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow b. The output is a morphism g: a \rightarrow b \otimes c corresponding to f under the cohom tensor adjunction.

##### 1.7-11 LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct

`‣ LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct`( a, c, f, t ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, t ).

The arguments are two objects a,c, a morphism f: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow b and an object t = b \otimes c. The output is a morphism g: a \rightarrow b \otimes c corresponding to f under the cohom tensor adjunction.

##### 1.7-12 LeftCoclosedMonoidalPreCoComposeMorphism

`‣ LeftCoclosedMonoidalPreCoComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,c), \mathrm{\underline{coHom}_\ell}(b,c) \otimes \mathrm{\underline{coHom}_\ell}(a,b) ).

The arguments are three objects a,b,c. The output is the precocomposition morphism \mathrm{LeftCoclosedMonoidalPreCoComposeMorphism}_{a,b,c}: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow \mathrm{\underline{coHom}_\ell}(b,c) \otimes \mathrm{\underline{coHom}_\ell}(a,b).

##### 1.7-13 LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects

`‣ LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{coHom}_\ell}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(b,c). The output is the precocomposition morphism \mathrm{LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow \mathrm{\underline{coHom}_\ell}(b,c) \otimes \mathrm{\underline{coHom}_\ell}(a,b).

##### 1.7-14 LeftCoclosedMonoidalPostCoComposeMorphism

`‣ LeftCoclosedMonoidalPostCoComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,c), \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(b,c) ).

The arguments are three objects a,b,c. The output is the postcocomposition morphism \mathrm{LeftCoclosedMonoidalPostCoComposeMorphism}_{a,b,c}: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(b,c).

##### 1.7-15 LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects

`‣ LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{coHom}_\ell}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}_\ell}(b,c) \otimes \mathrm{\underline{coHom}_\ell}(a,b). The output is the postcocomposition morphism \mathrm{LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}_\ell}(a,c) \rightarrow \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(b,c).

##### 1.7-16 LeftCoDualOnObjects

`‣ LeftCoDualOnObjects`( a ) | ( attribute )

Returns: an object

The argument is an object a. The output is its codual object a_{\vee}.

##### 1.7-17 LeftCoDualOnMorphisms

`‣ LeftCoDualOnMorphisms`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( b_{\vee}, a_{\vee} ).

The argument is a morphism \alpha: a \rightarrow b. The output is its codual morphism \alpha_{\vee}: b_{\vee} \rightarrow a_{\vee}.

##### 1.7-18 LeftCoDualOnMorphismsWithGivenLeftCoDuals

`‣ LeftCoDualOnMorphismsWithGivenLeftCoDuals`( s, alpha, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The argument is an object s = b_{\vee}, a morphism \alpha: a \rightarrow b, and an object r = a_{\vee}. The output is the dual morphism \alpha_{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.7-19 LeftCoclosedMonoidalEvaluationForLeftCoDual

`‣ LeftCoclosedMonoidalEvaluationForLeftCoDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( 1, a_{\vee} \otimes a ).

The argument is an object a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a.

##### 1.7-20 LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct

`‣ LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = 1, an object a, and an object r = a_{\vee} \otimes a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a.

##### 1.7-21 MorphismFromLeftCoBidual

`‣ MorphismFromLeftCoBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}((a_{\vee})_{\vee}, a).

The argument is an object a. The output is the morphism from the cobidual (a_{\vee})_{\vee} \rightarrow a.

##### 1.7-22 MorphismFromLeftCoBidualWithGivenLeftCoBidual

`‣ MorphismFromLeftCoBidualWithGivenLeftCoBidual`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(s, a).

The arguments are an object a, and an object s = (a_{\vee})_{\vee}. The output is the morphism from the cobidual (a_{\vee})_{\vee} \rightarrow a.

##### 1.7-23 LeftInternalCoHomTensorProductCompatibilityMorphism

`‣ LeftInternalCoHomTensorProductCompatibilityMorphism`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(a',b')).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{LeftInternalCoHomTensorProductCompatibilityMorphism}_{a,a',b,b'}: \mathrm{\underline{coHom}_\ell}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(a',b').

##### 1.7-24 LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

`‣ LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{coHom}_\ell}(a \otimes a', b \otimes b') and r = \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(a',b'). The output is the natural morphism \mathrm{LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}_\ell}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}_\ell}(a,b) \otimes \mathrm{\underline{coHom}_\ell}(a',b').

##### 1.7-25 LeftCoDualityTensorProductCompatibilityMorphism

`‣ LeftCoDualityTensorProductCompatibilityMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes b_{\vee} ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{LeftCoDualityTensorProductCompatibilityMorphism}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.

##### 1.7-26 LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects

`‣ LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = (a \otimes b)_{\vee}, two objects a,b, and an object r = a_{\vee} \otimes b_{\vee}. The output is the natural morphism \mathrm{LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects}_{a,b}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.

##### 1.7-27 MorphismFromLeftInternalCoHomToTensorProduct

`‣ MorphismFromLeftInternalCoHomToTensorProduct`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,b), b_{\vee} \otimes a ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromLeftInternalCoHomToTensorProduct}_{a,b}: \mathrm{\underline{coHom}_\ell}(a,b) \rightarrow b_{\vee} \otimes a.

##### 1.7-28 MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects

`‣ MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{coHom}_\ell}(a,b), two objects a,b, and an object r = b_{\vee} \otimes a. The output is the natural morphism \mathrm{MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}_\ell}(a,b) \rightarrow a \otimes b_{\vee}.

##### 1.7-29 IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit

`‣ IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a_{\vee}, \mathrm{\underline{coHom}_\ell}(1,a)).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit}_{a}: a_{\vee} \rightarrow \mathrm{\underline{coHom}_\ell}(1,a).

##### 1.7-30 IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject

`‣ IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}_\ell}(1,a), a_{\vee}).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject}_{a}: \mathrm{\underline{coHom}_\ell}(1,a) \rightarrow a_{\vee}.

##### 1.7-31 UniversalPropertyOfLeftCoDual

`‣ UniversalPropertyOfLeftCoDual`( t, a, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a_{\vee}, t).

The arguments are two objects t,a, and a morphism \alpha: 1 \rightarrow t \otimes a. The output is the morphism a_{\vee} \rightarrow t given by the universal property of a_{\vee}.

##### 1.7-32 LeftCoclosedMonoidalLambdaIntroduction

`‣ LeftCoclosedMonoidalLambdaIntroduction`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}_\ell}(a,b), 1 ).

The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism \mathrm{\underline{coHom}_\ell}(a,b) \rightarrow 1 under the cohom tensor adjunction.

##### 1.7-33 LeftCoclosedMonoidalLambdaElimination

`‣ LeftCoclosedMonoidalLambdaElimination`( a, b, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a,b).

The arguments are two objects a,b, and a morphism \alpha: \mathrm{\underline{coHom}_\ell}(a,b) \rightarrow 1. The output is a morphism a \rightarrow b corresponding to \alpha under the cohom tensor adjunction.

##### 1.7-34 IsomorphismFromObjectToLeftInternalCoHom

`‣ IsomorphismFromObjectToLeftInternalCoHom`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{coHom}_\ell}(a,1)).

The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}_\ell}(a,1).

##### 1.7-35 IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom

`‣ IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, r).

The argument is an object a, and an object r = \mathrm{\underline{coHom}_\ell}(a,1). The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}_\ell}(a,1).

##### 1.7-36 IsomorphismFromLeftInternalCoHomToObject

`‣ IsomorphismFromLeftInternalCoHomToObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}_\ell}(a,1), a).

The argument is an object a. The output is the natural isomorphism \mathrm{\underline{coHom}_\ell}(a,1) \rightarrow a.

##### 1.7-37 IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom

`‣ IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(s, a).

The argument is an object a, and an object s = \mathrm{\underline{coHom}_\ell}(a,1). The output is the natural isomorphism \mathrm{\underline{coHom}_\ell}(a,1) \rightarrow a.

#### 1.8 Coclosed Monoidal Categories

A monoidal category \mathbf{C} which has for each functor - \otimes b: \mathbf{C} \rightarrow \mathbf{C} a left adjoint (denoted by \mathrm{\underline{coHom}}(-,b)) is called a coclosed monoidal category.

If no operations involving coduals are installed manually, the codual objects will be derived as a_\vee \coloneqq \mathrm{\underline{coHom}}(1,a).

The corresponding GAP property is called `IsCoclosedMonoidalCategory`.

##### 1.8-1 InternalCoHomOnObjects

`‣ InternalCoHomOnObjects`( a, b ) | ( operation )

Returns: an object

The arguments are two objects a,b. The output is the internal cohom object \mathrm{\underline{coHom}}(a,b).

##### 1.8-2 InternalCoHomOnMorphisms

`‣ InternalCoHomOnMorphisms`( alpha, beta ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b'), \mathrm{\underline{coHom}}(a',b) )

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the internal cohom morphism \mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b).

##### 1.8-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms

`‣ InternalCoHomOnMorphismsWithGivenInternalCoHoms`( s, alpha, beta, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = \mathrm{\underline{coHom}}(a,b'), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{coHom}}(a',b). The output is the internal cohom morphism \mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b).

##### 1.8-4 CoclosedMonoidalRightEvaluationMorphism

`‣ CoclosedMonoidalRightEvaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, a \otimes \mathrm{\underline{coHom}}(b,a) ).

The arguments are two objects a, b. The output is the coclosed right evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow a \otimes \mathrm{\underline{coHom}}(b,a), i.e., the unit of the cohom tensor adjunction.

##### 1.8-5 CoclosedMonoidalRightEvaluationMorphismWithGivenRange

`‣ CoclosedMonoidalRightEvaluationMorphismWithGivenRange`( a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, r ).

The arguments are two objects a,b and an object r = a \otimes \mathrm{\underline{coHom}}(b,a). The output is the coclosed right evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow a \otimes \mathrm{\underline{coHom}}(b,a), i.e., the unit of the cohom tensor adjunction.

##### 1.8-6 CoclosedMonoidalRightCoevaluationMorphism

`‣ CoclosedMonoidalRightCoevaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes b, a), b ).

The arguments are two objects a,b. The output is the coclosed right coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, a) \rightarrow b, i.e., the counit of the cohom tensor adjunction.

##### 1.8-7 CoclosedMonoidalRightCoevaluationMorphismWithGivenSource

`‣ CoclosedMonoidalRightCoevaluationMorphismWithGivenSource`( a, b, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, b ).

The arguments are two objects a,b and an object s = \mathrm{\underline{coHom}}(a \otimes b, a). The output is the coclosed right coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, a) \rightarrow b, i.e., the unit of the cohom tensor adjunction.

##### 1.8-8 TensorProductToInternalCoHomRightAdjunctMorphism

`‣ TensorProductToInternalCoHomRightAdjunctMorphism`( b, c, g ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c ).

The arguments are two objects b,c and a morphism g: a \rightarrow b \otimes c. The output is a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c corresponding to g under the cohom tensor adjunction.

##### 1.8-9 TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom

`‣ TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom`( b, c, g, i ) | ( operation )

Returns: a morphism in \mathrm{Hom}( i, c ).

The arguments are two objects b,c, a morphism g: a \rightarrow b \otimes c and an object i = \mathrm{\underline{coHom}}(a,b). The output is a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c corresponding to g under the cohom tensor adjunction.

##### 1.8-10 InternalCoHomToTensorProductRightAdjunctMorphism

`‣ InternalCoHomToTensorProductRightAdjunctMorphism`( a, b, f ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, b \otimes c).

The arguments are two objects a,b and a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c. The output is a morphism g: a \rightarrow b \otimes c corresponding to f under the cohom tensor adjunction.

##### 1.8-11 InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct

`‣ InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct`( a, b, f, t ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, t ).

The arguments are two objects a,b, a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c and an object t = b \otimes c. The output is a morphism g: a \rightarrow t corresponding to f under the cohom tensor adjunction.

##### 1.8-12 CoclosedMonoidalLeftEvaluationMorphism

`‣ CoclosedMonoidalLeftEvaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, \mathrm{\underline{coHom}}(b,a) \otimes a ).

The arguments are two objects a, b. The output is the coclosed left evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow \mathrm{\underline{coHom}}(b,a) \otimes a, i.e., the unit of the cohom tensor adjunction.

##### 1.8-13 CoclosedMonoidalLeftEvaluationMorphismWithGivenRange

`‣ CoclosedMonoidalLeftEvaluationMorphismWithGivenRange`( a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b, r ).

The arguments are two objects a,b and an object r = \mathrm{\underline{coHom}}(b,a) \otimes a. The output is the coclosed left evaluation morphism \mathrm{coclev}_{a,b}: b \rightarrow \mathrm{\underline{coHom}}(b,a) \otimes a, i.e., the unit of the cohom tensor adjunction.

##### 1.8-14 CoclosedMonoidalLeftCoevaluationMorphism

`‣ CoclosedMonoidalLeftCoevaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(b \otimes a, a), b ).

The arguments are two objects a,b. The output is the coclosed left coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(b \otimes a, a) \rightarrow b, i.e., the counit of the cohom tensor adjunction.

##### 1.8-15 CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource

`‣ CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource`( a, b, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, b ).

The arguments are two objects a,b and an object s = \mathrm{\underline{coHom}}(b \otimes a, a). The output is the coclosed left coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(b \otimes a, a) \rightarrow b, i.e., the unit of the cohom tensor adjunction.

##### 1.8-16 TensorProductToInternalCoHomLeftAdjunctMorphism

`‣ TensorProductToInternalCoHomLeftAdjunctMorphism`( b, c, g ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), b ).

The arguments are two objects b,c and a morphism g: a \rightarrow b \otimes c. The output is a morphism f: \mathrm{\underline{coHom}}(a,c) \rightarrow b corresponding to g under the cohom tensor adjunction.

##### 1.8-17 TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom

`‣ TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom`( b, c, g, i ) | ( operation )

Returns: a morphism in \mathrm{Hom}( i, b ).

The arguments are two objects b,c, a morphism g: a \rightarrow b \otimes c and an object i = \mathrm{\underline{coHom}}(a,c). The output is a morphism f: \mathrm{\underline{coHom}}(a,c) \rightarrow b corresponding to g under the cohom tensor adjunction.

##### 1.8-18 InternalCoHomToTensorProductLeftAdjunctMorphism

`‣ InternalCoHomToTensorProductLeftAdjunctMorphism`( a, c, f ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, b \otimes c).

The arguments are two objects a,c and a morphism f: \mathrm{\underline{coHom}}(a,c) \rightarrow b. The output is a morphism g: a \rightarrow b \otimes c corresponding to f under the cohom tensor adjunction.

##### 1.8-19 InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct

`‣ InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct`( a, c, f, t ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, t ).

The arguments are two objects a,c, a morphism f: \mathrm{\underline{coHom}}(a,c) \rightarrow b and an object t = b \otimes c. The output is a morphism g: a \rightarrow t corresponding to f under the cohom tensor adjunction.

##### 1.8-20 MonoidalPreCoComposeMorphism

`‣ MonoidalPreCoComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b) ).

The arguments are three objects a,b,c. The output is the precocomposition morphism \mathrm{MonoidalPreCoComposeMorphism}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b).

##### 1.8-21 MonoidalPreCoComposeMorphismWithGivenObjects

`‣ MonoidalPreCoComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{coHom}}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c). The output is the precocomposition morphism \mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b).

##### 1.8-22 MonoidalPostCoComposeMorphism

`‣ MonoidalPostCoComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c) ).

The arguments are three objects a,b,c. The output is the postcocomposition morphism \mathrm{MonoidalPostCoComposeMorphism}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c).

##### 1.8-23 MonoidalPostCoComposeMorphismWithGivenObjects

`‣ MonoidalPostCoComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{coHom}}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b). The output is the postcocomposition morphism \mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c).

##### 1.8-24 CoDualOnObjects

`‣ CoDualOnObjects`( a ) | ( attribute )

Returns: an object

The argument is an object a. The output is its codual object a_{\vee}.

##### 1.8-25 CoDualOnMorphisms

`‣ CoDualOnMorphisms`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( b_{\vee}, a_{\vee} ).

The argument is a morphism \alpha: a \rightarrow b. The output is its codual morphism \alpha_{\vee}: b_{\vee} \rightarrow a_{\vee}.

##### 1.8-26 CoDualOnMorphismsWithGivenCoDuals

`‣ CoDualOnMorphismsWithGivenCoDuals`( s, alpha, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The argument is an object s = b_{\vee}, a morphism \alpha: a \rightarrow b, and an object r = a_{\vee}. The output is the dual morphism \alpha_{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.8-27 CoclosedEvaluationForCoDual

`‣ CoclosedEvaluationForCoDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( 1, a_{\vee} \otimes a ).

The argument is an object a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a.

##### 1.8-28 CoclosedEvaluationForCoDualWithGivenTensorProduct

`‣ CoclosedEvaluationForCoDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = 1, an object a, and an object r = a_{\vee} \otimes a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a.

##### 1.8-29 MorphismFromCoBidual

`‣ MorphismFromCoBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}((a_{\vee})_{\vee}, a).

The argument is an object a. The output is the morphism from the cobidual (a_{\vee})_{\vee} \rightarrow a.

##### 1.8-30 MorphismFromCoBidualWithGivenCoBidual

`‣ MorphismFromCoBidualWithGivenCoBidual`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(s, a).

The arguments are an object a, and an object s = (a_{\vee})_{\vee}. The output is the morphism from the cobidual (a_{\vee})_{\vee} \rightarrow a.

##### 1.8-31 InternalCoHomTensorProductCompatibilityMorphism

`‣ InternalCoHomTensorProductCompatibilityMorphism`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphism}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b').

##### 1.8-32 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

`‣ InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') and r = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'). The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b').

##### 1.8-33 CoDualityTensorProductCompatibilityMorphism

`‣ CoDualityTensorProductCompatibilityMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes b_{\vee} ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{CoDualityTensorProductCompatibilityMorphism}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.

##### 1.8-34 CoDualityTensorProductCompatibilityMorphismWithGivenObjects

`‣ CoDualityTensorProductCompatibilityMorphismWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = (a \otimes b)_{\vee}, two objects a,b, and an object r = a_{\vee} \otimes b_{\vee}. The output is the natural morphism \mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}_{a,b}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.

##### 1.8-35 MorphismFromInternalCoHomToTensorProduct

`‣ MorphismFromInternalCoHomToTensorProduct`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), b_{\vee} \otimes a ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromInternalCoHomToTensorProduct}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a.

##### 1.8-36 MorphismFromInternalCoHomToTensorProductWithGivenObjects

`‣ MorphismFromInternalCoHomToTensorProductWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r ).

The arguments are an object s = \mathrm{\underline{coHom}}(a,b), two objects a,b, and an object r = b_{\vee} \otimes a. The output is the natural morphism \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow a \otimes b_{\vee}.

##### 1.8-37 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit

`‣ IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a_{\vee}, \mathrm{\underline{coHom}}(1,a)).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}_{a}: a_{\vee} \rightarrow \mathrm{\underline{coHom}}(1,a).

##### 1.8-38 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject

`‣ IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}}(1,a), a_{\vee}).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}_{a}: \mathrm{\underline{coHom}}(1,a) \rightarrow a_{\vee}.

##### 1.8-39 UniversalPropertyOfCoDual

`‣ UniversalPropertyOfCoDual`( t, a, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a_{\vee}, t).

The arguments are two objects t,a, and a morphism \alpha: 1 \rightarrow t \otimes a. The output is the morphism a_{\vee} \rightarrow t given by the universal property of a_{\vee}.

##### 1.8-40 CoLambdaIntroduction

`‣ CoLambdaIntroduction`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), 1 ).

The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism \mathrm{\underline{coHom}}(a,b) \rightarrow 1 under the cohom tensor adjunction.

##### 1.8-41 CoLambdaElimination

`‣ CoLambdaElimination`( a, b, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a,b).

The arguments are two objects a,b, and a morphism \alpha: \mathrm{\underline{coHom}}(a,b) \rightarrow 1. The output is a morphism a \rightarrow b corresponding to \alpha under the cohom tensor adjunction.

##### 1.8-42 IsomorphismFromObjectToInternalCoHom

`‣ IsomorphismFromObjectToInternalCoHom`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1)).

The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}}(a,1).

##### 1.8-43 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

`‣ IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, r).

The argument is an object a, and an object r = \mathrm{\underline{coHom}}(a,1). The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}}(a,1).

##### 1.8-44 IsomorphismFromInternalCoHomToObject

`‣ IsomorphismFromInternalCoHomToObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a).

The argument is an object a. The output is the natural isomorphism \mathrm{\underline{coHom}}(a,1) \rightarrow a.

##### 1.8-45 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom

`‣ IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(s, a).

The argument is an object a, and an object s = \mathrm{\underline{coHom}}(a,1). The output is the natural isomorphism \mathrm{\underline{coHom}}(a,1) \rightarrow a.

#### 1.9 Symmetric Closed Monoidal Categories

A monoidal category \mathbf{C} which is symmetric and closed is called a symmetric closed monoidal category.

The corresponding GAP property is given by `IsSymmetricClosedMonoidalCategory`.

#### 1.10 Symmetric Coclosed Monoidal Categories

A monoidal category \mathbf{C} which is symmetric and coclosed is called a symmetric coclosed monoidal category.

The corresponding GAP property is given by `IsSymmetricCoclosedMonoidalCategory`.

#### 1.11 Rigid Symmetric Closed Monoidal Categories

A symmetric closed monoidal category \mathbf{C} satisfying

- the natural morphism

\mathrm{\underline{Hom}_\ell}(a, a') \otimes \mathrm{\underline{Hom}_\ell}(b, b') \rightarrow \mathrm{\underline{Hom}_\ell}(a \otimes b, a' \otimes b') is an isomorphism,

- the natural morphism

a \rightarrow \mathrm{\underline{Hom}_\ell}(\mathrm{\underline{Hom}_\ell}(a, 1), 1) is an isomorphism is called a rigid symmetric closed monoidal category.

If no operations involving the closed structure are installed manually, the internal hom objects will be derived as \mathrm{\underline{Hom}_\ell}(a,b) \coloneqq a^\vee \otimes b and, in particular, \mathrm{\underline{Hom}_\ell}(a,1) \coloneqq a^\vee \otimes 1.

The corresponding GAP property is given by `IsRigidSymmetricClosedMonoidalCategory`.

##### 1.11-1 IsomorphismFromTensorProductWithDualObjectToInternalHom

`‣ IsomorphismFromTensorProductWithDualObjectToInternalHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b).

##### 1.11-2 IsomorphismFromInternalHomToTensorProductWithDualObject

`‣ IsomorphismFromInternalHomToTensorProductWithDualObject`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b ).

The arguments are two objects a,b. The output is the inverse of \mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}, namely \mathrm{IsomorphismFromInternalHomToTensorProductWithDualObject}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b.

##### 1.11-3 MorphismFromInternalHomToTensorProduct

`‣ MorphismFromInternalHomToTensorProduct`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b ).

The arguments are two objects a,b. The output is the inverse of \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}, namely \mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b.

##### 1.11-4 MorphismFromInternalHomToTensorProductWithGivenObjects

`‣ MorphismFromInternalHomToTensorProductWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b ).

The arguments are an object s = \mathrm{\underline{Hom}}(a,b), two objects a,b, and an object r = a^{\vee} \otimes b. The output is the inverse of \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}, namely \mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b.

##### 1.11-5 TensorProductInternalHomCompatibilityMorphismInverse

`‣ TensorProductInternalHomCompatibilityMorphismInverse`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') ).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b').

##### 1.11-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects

`‣ TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') and r = \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'). The output is the natural morphism \mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b').

##### 1.11-7 CoevaluationForDual

`‣ CoevaluationForDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,a \otimes a^{\vee}).

The argument is an object a. The output is the coevaluation morphism \mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}.

##### 1.11-8 CoevaluationForDualWithGivenTensorProduct

`‣ CoevaluationForDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(1,a \otimes a^{\vee}).

The arguments are an object s = 1, an object a, and an object r = a \otimes a^{\vee}. The output is the coevaluation morphism \mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}.

##### 1.11-9 TraceMap

`‣ TraceMap`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,1).

The argument is an endomorphism \alpha: a \rightarrow a. The output is the trace morphism \mathrm{trace}_{\alpha}: 1 \rightarrow 1.

##### 1.11-10 RankMorphism

`‣ RankMorphism`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,1).

The argument is an object a. The output is the rank morphism \mathrm{rank}_a: 1 \rightarrow 1.

##### 1.11-11 MorphismFromBidual

`‣ MorphismFromBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}((a^{\vee})^{\vee},a).

The argument is an object a. The output is the inverse of the morphism to the bidual (a^{\vee})^{\vee} \rightarrow a.

##### 1.11-12 MorphismFromBidualWithGivenBidual

`‣ MorphismFromBidualWithGivenBidual`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}((a^{\vee})^{\vee},a).

The argument is an object a, and an object s = (a^{\vee})^{\vee}. The output is the inverse of the morphism to the bidual (a^{\vee})^{\vee} \rightarrow a.

#### 1.12 Rigid Symmetric Coclosed Monoidal Categories

A symmetric coclosed monoidal category \mathbf{C} satisfying

- the natural morphism

\mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a, b) \otimes \mathrm{\underline{coHom}}(a', b') is an isomorphism,

- the natural morphism

\mathrm{\underline{coHom}}(1, \mathrm{\underline{coHom}}(1, a)) \rightarrow a is an isomorphism is called a rigid symmetric coclosed monoidal category.

If no operations involving the coclosed structure are installed manually, the internal cohom objects will be derived as \mathrm{\underline{coHom}}(a,b) \coloneqq a \otimes b_\vee and, in particular, \mathrm{\underline{coHom}}(1,a) \coloneqq 1 \otimes a_\vee.

The corresponding GAP property is given by `IsRigidSymmetricCoclosedMonoidalCategory`.

##### 1.12-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject

`‣ IsomorphismFromInternalCoHomToTensorProductWithCoDualObject`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), b_{\vee} \otimes a ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObjectWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a.

##### 1.12-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom

`‣ IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a).

The arguments are two objects a,b. The output is the inverse of \mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}, namely \mathrm{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).

##### 1.12-3 MorphismFromTensorProductToInternalCoHom

`‣ MorphismFromTensorProductToInternalCoHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a) ).

The arguments are two objects a,b. The output is the inverse of \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}, namely \mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).

##### 1.12-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects

`‣ MorphismFromTensorProductToInternalCoHomWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a).

The arguments are an object s_{\vee} = a \otimes b, two objects a,b, and an object r = \mathrm{\underline{coHom}}(b,a). The output is the inverse of \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}, namely \mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).

##### 1.12-5 InternalCoHomTensorProductCompatibilityMorphismInverse

`‣ InternalCoHomTensorProductCompatibilityMorphismInverse`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'), \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' ).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b').

##### 1.12-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects

`‣ InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'), \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') and r = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'). The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b').

##### 1.12-7 CoclosedCoevaluationForCoDual

`‣ CoclosedCoevaluationForCoDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a \otimes a_{\vee}, 1).

The argument is an object a. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1.

##### 1.12-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct

`‣ CoclosedCoevaluationForCoDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes a_{\vee}, 1).

The arguments are an object s = a \otimes a_{\vee}, an object a, and an object r = 1. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1.

##### 1.12-9 CoTraceMap

`‣ CoTraceMap`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,1).

The argument is an endomorphism \alpha: a \rightarrow a. The output is the cotrace morphism \mathrm{cotrace}_{\alpha}: 1 \rightarrow 1.

##### 1.12-10 CoRankMorphism

`‣ CoRankMorphism`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,1).

The argument is an object a. The output is the corank morphism \mathrm{corank}_a: 1 \rightarrow 1.

##### 1.12-11 MorphismToCoBidual

`‣ MorphismToCoBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, (a_{\vee})_{\vee}).

The argument is an object a. The output is the inverse of the morphism from the cobidual a \rightarrow (a_{\vee})_{\vee}.

##### 1.12-12 MorphismToCoBidualWithGivenCoBidual

`‣ MorphismToCoBidualWithGivenCoBidual`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a,(a_{\vee})_{\vee}).

The argument is an object a, and an object r = (a_{\vee})_{\vee}. The output is the inverse of the morphism from the cobidual a \rightarrow (a_{\vee})_{\vee}.

#### 1.13 Convenience Methods

##### 1.13-1 InternalHom

`‣ InternalHom`( a, b ) | ( operation )

Returns: a cell

This is a convenience method. The arguments are two cells a,b. The output is the internal hom cell. If a,b are two CAP objects the output is the internal Hom object \mathrm{\underline{Hom}}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal hom on morphisms, where any object is replaced by its identity morphism.

##### 1.13-2 InternalCoHom

`‣ InternalCoHom`( a, b ) | ( operation )

Returns: a cell

This is a convenience method. The arguments are two cells a,b. The output is the internal cohom cell. If a,b are two CAP objects the output is the internal cohom object \mathrm{\underline{coHom}}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal cohom on morphisms, where any object is replaced by its identity morphism.

##### 1.13-3 LeftInternalHom

`‣ LeftInternalHom`( a, b ) | ( operation )

Returns: a cell

This is a convenience method. The arguments are two cells a,b. The output is the internal hom cell. If a,b are two CAP objects the output is the internal Hom object \mathrm{\underline{Hom}_\ell}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal hom on morphisms, where any object is replaced by its identity morphism.

##### 1.13-4 LeftInternalCoHom

`‣ LeftInternalCoHom`( a, b ) | ( operation )

Returns: a cell

This is a convenience method. The arguments are two cells a,b. The output is the internal cohom cell. If a,b are two CAP objects the output is the internal cohom object \mathrm{\underline{coHom}_\ell}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal cohom on morphisms, where any object is replaced by its identity morphism.

#### 1.14 Add-methods

##### 1.14-1 AddLeftDistributivityExpanding

`‣ AddLeftDistributivityExpanding`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDistributivityExpanding`. F: ( a, L ) \mapsto \mathtt{LeftDistributivityExpanding}(a, L).

##### 1.14-2 AddLeftDistributivityExpandingWithGivenObjects

`‣ AddLeftDistributivityExpandingWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDistributivityExpandingWithGivenObjects`. F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityExpandingWithGivenObjects}(s, a, L, r).

##### 1.14-3 AddLeftDistributivityFactoring

`‣ AddLeftDistributivityFactoring`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDistributivityFactoring`. F: ( a, L ) \mapsto \mathtt{LeftDistributivityFactoring}(a, L).

##### 1.14-4 AddLeftDistributivityFactoringWithGivenObjects

`‣ AddLeftDistributivityFactoringWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDistributivityFactoringWithGivenObjects`. F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityFactoringWithGivenObjects}(s, a, L, r).

##### 1.14-5 AddRightDistributivityExpanding

`‣ AddRightDistributivityExpanding`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightDistributivityExpanding`. F: ( L, a ) \mapsto \mathtt{RightDistributivityExpanding}(L, a).

##### 1.14-6 AddRightDistributivityExpandingWithGivenObjects

`‣ AddRightDistributivityExpandingWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightDistributivityExpandingWithGivenObjects`. F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityExpandingWithGivenObjects}(s, L, a, r).

##### 1.14-7 AddRightDistributivityFactoring

`‣ AddRightDistributivityFactoring`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightDistributivityFactoring`. F: ( L, a ) \mapsto \mathtt{RightDistributivityFactoring}(L, a).

##### 1.14-8 AddRightDistributivityFactoringWithGivenObjects

`‣ AddRightDistributivityFactoringWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightDistributivityFactoringWithGivenObjects`. F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityFactoringWithGivenObjects}(s, L, a, r).

##### 1.14-9 AddBraiding

`‣ AddBraiding`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `Braiding`. F: ( a, b ) \mapsto \mathtt{Braiding}(a, b).

##### 1.14-10 AddBraidingInverse

`‣ AddBraidingInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `BraidingInverse`. F: ( a, b ) \mapsto \mathtt{BraidingInverse}(a, b).

##### 1.14-11 AddBraidingInverseWithGivenTensorProducts

`‣ AddBraidingInverseWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `BraidingInverseWithGivenTensorProducts`. F: ( s, a, b, r ) \mapsto \mathtt{BraidingInverseWithGivenTensorProducts}(s, a, b, r).

##### 1.14-12 AddBraidingWithGivenTensorProducts

`‣ AddBraidingWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `BraidingWithGivenTensorProducts`. F: ( s, a, b, r ) \mapsto \mathtt{BraidingWithGivenTensorProducts}(s, a, b, r).

##### 1.14-13 AddClosedMonoidalLeftCoevaluationMorphism

`‣ AddClosedMonoidalLeftCoevaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `ClosedMonoidalLeftCoevaluationMorphism`. F: ( a, b ) \mapsto \mathtt{ClosedMonoidalLeftCoevaluationMorphism}(a, b).

##### 1.14-14 AddClosedMonoidalLeftCoevaluationMorphismWithGivenRange

`‣ AddClosedMonoidalLeftCoevaluationMorphismWithGivenRange`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `ClosedMonoidalLeftCoevaluationMorphismWithGivenRange`. F: ( a, b, r ) \mapsto \mathtt{ClosedMonoidalLeftCoevaluationMorphismWithGivenRange}(a, b, r).

##### 1.14-15 AddClosedMonoidalLeftEvaluationMorphism

`‣ AddClosedMonoidalLeftEvaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `ClosedMonoidalLeftEvaluationMorphism`. F: ( a, b ) \mapsto \mathtt{ClosedMonoidalLeftEvaluationMorphism}(a, b).

##### 1.14-16 AddClosedMonoidalLeftEvaluationMorphismWithGivenSource

`‣ AddClosedMonoidalLeftEvaluationMorphismWithGivenSource`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `ClosedMonoidalLeftEvaluationMorphismWithGivenSource`. F: ( a, b, s ) \mapsto \mathtt{ClosedMonoidalLeftEvaluationMorphismWithGivenSource}(a, b, s).

##### 1.14-17 AddClosedMonoidalRightCoevaluationMorphism

`‣ AddClosedMonoidalRightCoevaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `ClosedMonoidalRightCoevaluationMorphism`. F: ( a, b ) \mapsto \mathtt{ClosedMonoidalRightCoevaluationMorphism}(a, b).

##### 1.14-18 AddClosedMonoidalRightCoevaluationMorphismWithGivenRange

`‣ AddClosedMonoidalRightCoevaluationMorphismWithGivenRange`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `ClosedMonoidalRightCoevaluationMorphismWithGivenRange`. F: ( a, b, r ) \mapsto \mathtt{ClosedMonoidalRightCoevaluationMorphismWithGivenRange}(a, b, r).

##### 1.14-19 AddClosedMonoidalRightEvaluationMorphism

`‣ AddClosedMonoidalRightEvaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `ClosedMonoidalRightEvaluationMorphism`. F: ( a, b ) \mapsto \mathtt{ClosedMonoidalRightEvaluationMorphism}(a, b).

##### 1.14-20 AddClosedMonoidalRightEvaluationMorphismWithGivenSource

`‣ AddClosedMonoidalRightEvaluationMorphismWithGivenSource`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `ClosedMonoidalRightEvaluationMorphismWithGivenSource`. F: ( a, b, s ) \mapsto \mathtt{ClosedMonoidalRightEvaluationMorphismWithGivenSource}(a, b, s).

##### 1.14-21 AddDualOnMorphisms

`‣ AddDualOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `DualOnMorphisms`. F: ( alpha ) \mapsto \mathtt{DualOnMorphisms}(alpha).

##### 1.14-22 AddDualOnMorphismsWithGivenDuals

`‣ AddDualOnMorphismsWithGivenDuals`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `DualOnMorphismsWithGivenDuals`. F: ( s, alpha, r ) \mapsto \mathtt{DualOnMorphismsWithGivenDuals}(s, alpha, r).

##### 1.14-23 AddDualOnObjects

`‣ AddDualOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `DualOnObjects`. F: ( a ) \mapsto \mathtt{DualOnObjects}(a).

##### 1.14-24 AddEvaluationForDual

`‣ AddEvaluationForDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `EvaluationForDual`. F: ( a ) \mapsto \mathtt{EvaluationForDual}(a).

##### 1.14-25 AddEvaluationForDualWithGivenTensorProduct

`‣ AddEvaluationForDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `EvaluationForDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{EvaluationForDualWithGivenTensorProduct}(s, a, r).

##### 1.14-26 AddInternalHomOnMorphisms

`‣ AddInternalHomOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomOnMorphisms`. F: ( alpha, beta ) \mapsto \mathtt{InternalHomOnMorphisms}(alpha, beta).

##### 1.14-27 AddInternalHomOnMorphismsWithGivenInternalHoms

`‣ AddInternalHomOnMorphismsWithGivenInternalHoms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomOnMorphismsWithGivenInternalHoms`. F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalHomOnMorphismsWithGivenInternalHoms}(s, alpha, beta, r).

##### 1.14-28 AddInternalHomOnObjects

`‣ AddInternalHomOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomOnObjects`. F: ( a, b ) \mapsto \mathtt{InternalHomOnObjects}(a, b).

##### 1.14-29 AddInternalHomToTensorProductLeftAdjunctMorphism

`‣ AddInternalHomToTensorProductLeftAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductLeftAdjunctMorphism`. F: ( b, c, g ) \mapsto \mathtt{InternalHomToTensorProductLeftAdjunctMorphism}(b, c, g).

##### 1.14-30 AddInternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct

`‣ AddInternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct`. F: ( b, c, g, s ) \mapsto \mathtt{InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct}(b, c, g, s).

##### 1.14-31 AddInternalHomToTensorProductLeftAdjunctionIsomorphism

`‣ AddInternalHomToTensorProductLeftAdjunctionIsomorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductLeftAdjunctionIsomorphism`. F: ( a, b, c ) \mapsto \mathtt{InternalHomToTensorProductLeftAdjunctionIsomorphism}(a, b, c).

##### 1.14-32 AddInternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects

`‣ AddInternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{InternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-33 AddInternalHomToTensorProductRightAdjunctMorphism

`‣ AddInternalHomToTensorProductRightAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductRightAdjunctMorphism`. F: ( a, c, g ) \mapsto \mathtt{InternalHomToTensorProductRightAdjunctMorphism}(a, c, g).

##### 1.14-34 AddInternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct

`‣ AddInternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct`. F: ( a, c, g, s ) \mapsto \mathtt{InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct}(a, c, g, s).

##### 1.14-35 AddInternalHomToTensorProductRightAdjunctionIsomorphism

`‣ AddInternalHomToTensorProductRightAdjunctionIsomorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductRightAdjunctionIsomorphism`. F: ( a, b, c ) \mapsto \mathtt{InternalHomToTensorProductRightAdjunctionIsomorphism}(a, b, c).

##### 1.14-36 AddInternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects

`‣ AddInternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{InternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-37 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit

`‣ AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromDualObjectToInternalHomIntoTensorUnit`. F: ( a ) \mapsto \mathtt{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}(a).

##### 1.14-38 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject

`‣ AddIsomorphismFromInternalHomIntoTensorUnitToDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalHomIntoTensorUnitToDualObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomIntoTensorUnitToDualObject}(a).

##### 1.14-39 AddIsomorphismFromInternalHomToObject

`‣ AddIsomorphismFromInternalHomToObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalHomToObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomToObject}(a).

##### 1.14-40 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom

`‣ AddIsomorphismFromInternalHomToObjectWithGivenInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalHomToObjectWithGivenInternalHom`. F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalHomToObjectWithGivenInternalHom}(a, s).

##### 1.14-41 AddIsomorphismFromObjectToInternalHom

`‣ AddIsomorphismFromObjectToInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToInternalHom`. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalHom}(a).

##### 1.14-42 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom

`‣ AddIsomorphismFromObjectToInternalHomWithGivenInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToInternalHomWithGivenInternalHom`. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalHomWithGivenInternalHom}(a, r).

##### 1.14-43 AddLambdaElimination

`‣ AddLambdaElimination`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LambdaElimination`. F: ( a, b, alpha ) \mapsto \mathtt{LambdaElimination}(a, b, alpha).

##### 1.14-44 AddLambdaIntroduction

`‣ AddLambdaIntroduction`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LambdaIntroduction`. F: ( alpha ) \mapsto \mathtt{LambdaIntroduction}(alpha).

##### 1.14-45 AddMonoidalPostComposeMorphism

`‣ AddMonoidalPostComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPostComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{MonoidalPostComposeMorphism}(a, b, c).

##### 1.14-46 AddMonoidalPostComposeMorphismWithGivenObjects

`‣ AddMonoidalPostComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPostComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-47 AddMonoidalPreComposeMorphism

`‣ AddMonoidalPreComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPreComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{MonoidalPreComposeMorphism}(a, b, c).

##### 1.14-48 AddMonoidalPreComposeMorphismWithGivenObjects

`‣ AddMonoidalPreComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPreComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-49 AddMorphismFromTensorProductToInternalHom

`‣ AddMorphismFromTensorProductToInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToInternalHom`. F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalHom}(a, b).

##### 1.14-50 AddMorphismFromTensorProductToInternalHomWithGivenObjects

`‣ AddMorphismFromTensorProductToInternalHomWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToInternalHomWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalHomWithGivenObjects}(s, a, b, r).

##### 1.14-51 AddMorphismToBidual

`‣ AddMorphismToBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToBidual`. F: ( a ) \mapsto \mathtt{MorphismToBidual}(a).

##### 1.14-52 AddMorphismToBidualWithGivenBidual

`‣ AddMorphismToBidualWithGivenBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToBidualWithGivenBidual`. F: ( a, r ) \mapsto \mathtt{MorphismToBidualWithGivenBidual}(a, r).

##### 1.14-53 AddTensorProductDualityCompatibilityMorphism

`‣ AddTensorProductDualityCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductDualityCompatibilityMorphism`. F: ( a, b ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphism}(a, b).

##### 1.14-54 AddTensorProductDualityCompatibilityMorphismWithGivenObjects

`‣ AddTensorProductDualityCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductDualityCompatibilityMorphismWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r).

##### 1.14-55 AddTensorProductInternalHomCompatibilityMorphism

`‣ AddTensorProductInternalHomCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductInternalHomCompatibilityMorphism`. F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphism}(list).

##### 1.14-56 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects

`‣ AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductInternalHomCompatibilityMorphismWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}(source, list, range).

##### 1.14-57 AddTensorProductToInternalHomLeftAdjunctMorphism

`‣ AddTensorProductToInternalHomLeftAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomLeftAdjunctMorphism`. F: ( a, b, f ) \mapsto \mathtt{TensorProductToInternalHomLeftAdjunctMorphism}(a, b, f).

##### 1.14-58 AddTensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom

`‣ AddTensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom`. F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom}(a, b, f, i).

##### 1.14-59 AddTensorProductToInternalHomLeftAdjunctionIsomorphism

`‣ AddTensorProductToInternalHomLeftAdjunctionIsomorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomLeftAdjunctionIsomorphism`. F: ( a, b, c ) \mapsto \mathtt{TensorProductToInternalHomLeftAdjunctionIsomorphism}(a, b, c).

##### 1.14-60 AddTensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects

`‣ AddTensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{TensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-61 AddTensorProductToInternalHomRightAdjunctMorphism

`‣ AddTensorProductToInternalHomRightAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomRightAdjunctMorphism`. F: ( a, b, f ) \mapsto \mathtt{TensorProductToInternalHomRightAdjunctMorphism}(a, b, f).

##### 1.14-62 AddTensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom

`‣ AddTensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom`. F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom}(a, b, f, i).

##### 1.14-63 AddTensorProductToInternalHomRightAdjunctionIsomorphism

`‣ AddTensorProductToInternalHomRightAdjunctionIsomorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomRightAdjunctionIsomorphism`. F: ( a, b, c ) \mapsto \mathtt{TensorProductToInternalHomRightAdjunctionIsomorphism}(a, b, c).

##### 1.14-64 AddTensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects

`‣ AddTensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{TensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-65 AddUniversalPropertyOfDual

`‣ AddUniversalPropertyOfDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `UniversalPropertyOfDual`. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfDual}(t, a, alpha).

##### 1.14-66 AddCoDualOnMorphisms

`‣ AddCoDualOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualOnMorphisms`. F: ( alpha ) \mapsto \mathtt{CoDualOnMorphisms}(alpha).

##### 1.14-67 AddCoDualOnMorphismsWithGivenCoDuals

`‣ AddCoDualOnMorphismsWithGivenCoDuals`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualOnMorphismsWithGivenCoDuals`. F: ( s, alpha, r ) \mapsto \mathtt{CoDualOnMorphismsWithGivenCoDuals}(s, alpha, r).

##### 1.14-68 AddCoDualOnObjects

`‣ AddCoDualOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualOnObjects`. F: ( a ) \mapsto \mathtt{CoDualOnObjects}(a).

##### 1.14-69 AddCoDualityTensorProductCompatibilityMorphism

`‣ AddCoDualityTensorProductCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualityTensorProductCompatibilityMorphism`. F: ( a, b ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphism}(a, b).

##### 1.14-70 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects

`‣ AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualityTensorProductCompatibilityMorphismWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}(s, a, b, r).

##### 1.14-71 AddCoLambdaElimination

`‣ AddCoLambdaElimination`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoLambdaElimination`. F: ( a, b, alpha ) \mapsto \mathtt{CoLambdaElimination}(a, b, alpha).

##### 1.14-72 AddCoLambdaIntroduction

`‣ AddCoLambdaIntroduction`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoLambdaIntroduction`. F: ( alpha ) \mapsto \mathtt{CoLambdaIntroduction}(alpha).

##### 1.14-73 AddCoclosedEvaluationForCoDual

`‣ AddCoclosedEvaluationForCoDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedEvaluationForCoDual`. F: ( a ) \mapsto \mathtt{CoclosedEvaluationForCoDual}(a).

##### 1.14-74 AddCoclosedEvaluationForCoDualWithGivenTensorProduct

`‣ AddCoclosedEvaluationForCoDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedEvaluationForCoDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{CoclosedEvaluationForCoDualWithGivenTensorProduct}(s, a, r).

##### 1.14-75 AddCoclosedMonoidalLeftCoevaluationMorphism

`‣ AddCoclosedMonoidalLeftCoevaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedMonoidalLeftCoevaluationMorphism`. F: ( a, b ) \mapsto \mathtt{CoclosedMonoidalLeftCoevaluationMorphism}(a, b).

##### 1.14-76 AddCoclosedMonoidalLeftCoevaluationMorphismWithGivenSource

`‣ AddCoclosedMonoidalLeftCoevaluationMorphismWithGivenSource`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource`. F: ( a, b, s ) \mapsto \mathtt{CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource}(a, b, s).

##### 1.14-77 AddCoclosedMonoidalLeftEvaluationMorphism

`‣ AddCoclosedMonoidalLeftEvaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedMonoidalLeftEvaluationMorphism`. F: ( a, b ) \mapsto \mathtt{CoclosedMonoidalLeftEvaluationMorphism}(a, b).

##### 1.14-78 AddCoclosedMonoidalLeftEvaluationMorphismWithGivenRange

`‣ AddCoclosedMonoidalLeftEvaluationMorphismWithGivenRange`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedMonoidalLeftEvaluationMorphismWithGivenRange`. F: ( a, b, r ) \mapsto \mathtt{CoclosedMonoidalLeftEvaluationMorphismWithGivenRange}(a, b, r).

##### 1.14-79 AddCoclosedMonoidalRightCoevaluationMorphism

`‣ AddCoclosedMonoidalRightCoevaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedMonoidalRightCoevaluationMorphism`. F: ( a, b ) \mapsto \mathtt{CoclosedMonoidalRightCoevaluationMorphism}(a, b).

##### 1.14-80 AddCoclosedMonoidalRightCoevaluationMorphismWithGivenSource

`‣ AddCoclosedMonoidalRightCoevaluationMorphismWithGivenSource`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedMonoidalRightCoevaluationMorphismWithGivenSource`. F: ( a, b, s ) \mapsto \mathtt{CoclosedMonoidalRightCoevaluationMorphismWithGivenSource}(a, b, s).

##### 1.14-81 AddCoclosedMonoidalRightEvaluationMorphism

`‣ AddCoclosedMonoidalRightEvaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedMonoidalRightEvaluationMorphism`. F: ( a, b ) \mapsto \mathtt{CoclosedMonoidalRightEvaluationMorphism}(a, b).

##### 1.14-82 AddCoclosedMonoidalRightEvaluationMorphismWithGivenRange

`‣ AddCoclosedMonoidalRightEvaluationMorphismWithGivenRange`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedMonoidalRightEvaluationMorphismWithGivenRange`. F: ( a, b, r ) \mapsto \mathtt{CoclosedMonoidalRightEvaluationMorphismWithGivenRange}(a, b, r).

##### 1.14-83 AddInternalCoHomOnMorphisms

`‣ AddInternalCoHomOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomOnMorphisms`. F: ( alpha, beta ) \mapsto \mathtt{InternalCoHomOnMorphisms}(alpha, beta).

##### 1.14-84 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms

`‣ AddInternalCoHomOnMorphismsWithGivenInternalCoHoms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomOnMorphismsWithGivenInternalCoHoms`. F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalCoHomOnMorphismsWithGivenInternalCoHoms}(s, alpha, beta, r).

##### 1.14-85 AddInternalCoHomOnObjects

`‣ AddInternalCoHomOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomOnObjects`. F: ( a, b ) \mapsto \mathtt{InternalCoHomOnObjects}(a, b).

##### 1.14-86 AddInternalCoHomTensorProductCompatibilityMorphism

`‣ AddInternalCoHomTensorProductCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomTensorProductCompatibilityMorphism`. F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphism}(list).

##### 1.14-87 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

`‣ AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}(source, list, range).

##### 1.14-88 AddInternalCoHomToTensorProductLeftAdjunctMorphism

`‣ AddInternalCoHomToTensorProductLeftAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomToTensorProductLeftAdjunctMorphism`. F: ( a, c, f ) \mapsto \mathtt{InternalCoHomToTensorProductLeftAdjunctMorphism}(a, c, f).

##### 1.14-89 AddInternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct

`‣ AddInternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct`. F: ( a, c, f, t ) \mapsto \mathtt{InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct}(a, c, f, t).

##### 1.14-90 AddInternalCoHomToTensorProductRightAdjunctMorphism

`‣ AddInternalCoHomToTensorProductRightAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomToTensorProductRightAdjunctMorphism`. F: ( a, b, f ) \mapsto \mathtt{InternalCoHomToTensorProductRightAdjunctMorphism}(a, b, f).

##### 1.14-91 AddInternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct

`‣ AddInternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct`. F: ( a, b, f, t ) \mapsto \mathtt{InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct}(a, b, f, t).

##### 1.14-92 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit

`‣ AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit`. F: ( a ) \mapsto \mathtt{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}(a).

##### 1.14-93 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject

`‣ AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}(a).

##### 1.14-94 AddIsomorphismFromInternalCoHomToObject

`‣ AddIsomorphismFromInternalCoHomToObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalCoHomToObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObject}(a).

##### 1.14-95 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom

`‣ AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom`. F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom}(a, s).

##### 1.14-96 AddIsomorphismFromObjectToInternalCoHom

`‣ AddIsomorphismFromObjectToInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToInternalCoHom`. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHom}(a).

##### 1.14-97 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

`‣ AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom`. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom}(a, r).

##### 1.14-98 AddMonoidalPostCoComposeMorphism

`‣ AddMonoidalPostCoComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPostCoComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{MonoidalPostCoComposeMorphism}(a, b, c).

##### 1.14-99 AddMonoidalPostCoComposeMorphismWithGivenObjects

`‣ AddMonoidalPostCoComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPostCoComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostCoComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-100 AddMonoidalPreCoComposeMorphism

`‣ AddMonoidalPreCoComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPreCoComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{MonoidalPreCoComposeMorphism}(a, b, c).

##### 1.14-101 AddMonoidalPreCoComposeMorphismWithGivenObjects

`‣ AddMonoidalPreCoComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPreCoComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreCoComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-102 AddMorphismFromCoBidual

`‣ AddMorphismFromCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromCoBidual`. F: ( a ) \mapsto \mathtt{MorphismFromCoBidual}(a).

##### 1.14-103 AddMorphismFromCoBidualWithGivenCoBidual

`‣ AddMorphismFromCoBidualWithGivenCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromCoBidualWithGivenCoBidual`. F: ( a, s ) \mapsto \mathtt{MorphismFromCoBidualWithGivenCoBidual}(a, s).

##### 1.14-104 AddMorphismFromInternalCoHomToTensorProduct

`‣ AddMorphismFromInternalCoHomToTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromInternalCoHomToTensorProduct`. F: ( a, b ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProduct}(a, b).

##### 1.14-105 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects

`‣ AddMorphismFromInternalCoHomToTensorProductWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromInternalCoHomToTensorProductWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProductWithGivenObjects}(s, a, b, r).

##### 1.14-106 AddTensorProductToInternalCoHomLeftAdjunctMorphism

`‣ AddTensorProductToInternalCoHomLeftAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalCoHomLeftAdjunctMorphism`. F: ( b, c, g ) \mapsto \mathtt{TensorProductToInternalCoHomLeftAdjunctMorphism}(b, c, g).

##### 1.14-107 AddTensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom

`‣ AddTensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom`. F: ( b, c, g, i ) \mapsto \mathtt{TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom}(b, c, g, i).

##### 1.14-108 AddTensorProductToInternalCoHomRightAdjunctMorphism

`‣ AddTensorProductToInternalCoHomRightAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalCoHomRightAdjunctMorphism`. F: ( b, c, g ) \mapsto \mathtt{TensorProductToInternalCoHomRightAdjunctMorphism}(b, c, g).

##### 1.14-109 AddTensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom

`‣ AddTensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom`. F: ( b, c, g, i ) \mapsto \mathtt{TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom}(b, c, g, i).

##### 1.14-110 AddUniversalPropertyOfCoDual

`‣ AddUniversalPropertyOfCoDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `UniversalPropertyOfCoDual`. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfCoDual}(t, a, alpha).

##### 1.14-111 AddIsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit

`‣ AddIsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit`. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit}(a).

##### 1.14-112 AddIsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject

`‣ AddIsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject}(a).

##### 1.14-113 AddIsomorphismFromLeftInternalHomToObject

`‣ AddIsomorphismFromLeftInternalHomToObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromLeftInternalHomToObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftInternalHomToObject}(a).

##### 1.14-114 AddIsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom

`‣ AddIsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom`. F: ( a, s ) \mapsto \mathtt{IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom}(a, s).

##### 1.14-115 AddIsomorphismFromObjectToLeftInternalHom

`‣ AddIsomorphismFromObjectToLeftInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToLeftInternalHom`. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToLeftInternalHom}(a).

##### 1.14-116 AddIsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom

`‣ AddIsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom`. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom}(a, r).

##### 1.14-117 AddLeftClosedMonoidalCoevaluationMorphism

`‣ AddLeftClosedMonoidalCoevaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalCoevaluationMorphism`. F: ( a, b ) \mapsto \mathtt{LeftClosedMonoidalCoevaluationMorphism}(a, b).

##### 1.14-118 AddLeftClosedMonoidalCoevaluationMorphismWithGivenRange

`‣ AddLeftClosedMonoidalCoevaluationMorphismWithGivenRange`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalCoevaluationMorphismWithGivenRange`. F: ( a, b, r ) \mapsto \mathtt{LeftClosedMonoidalCoevaluationMorphismWithGivenRange}(a, b, r).

##### 1.14-119 AddLeftClosedMonoidalEvaluationForLeftDual

`‣ AddLeftClosedMonoidalEvaluationForLeftDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalEvaluationForLeftDual`. F: ( a ) \mapsto \mathtt{LeftClosedMonoidalEvaluationForLeftDual}(a).

##### 1.14-120 AddLeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct

`‣ AddLeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct}(s, a, r).

##### 1.14-121 AddLeftClosedMonoidalEvaluationMorphism

`‣ AddLeftClosedMonoidalEvaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalEvaluationMorphism`. F: ( a, b ) \mapsto \mathtt{LeftClosedMonoidalEvaluationMorphism}(a, b).

##### 1.14-122 AddLeftClosedMonoidalEvaluationMorphismWithGivenSource

`‣ AddLeftClosedMonoidalEvaluationMorphismWithGivenSource`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalEvaluationMorphismWithGivenSource`. F: ( a, b, s ) \mapsto \mathtt{LeftClosedMonoidalEvaluationMorphismWithGivenSource}(a, b, s).

##### 1.14-123 AddLeftClosedMonoidalLambdaElimination

`‣ AddLeftClosedMonoidalLambdaElimination`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalLambdaElimination`. F: ( a, b, alpha ) \mapsto \mathtt{LeftClosedMonoidalLambdaElimination}(a, b, alpha).

##### 1.14-124 AddLeftClosedMonoidalLambdaIntroduction

`‣ AddLeftClosedMonoidalLambdaIntroduction`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalLambdaIntroduction`. F: ( alpha ) \mapsto \mathtt{LeftClosedMonoidalLambdaIntroduction}(alpha).

##### 1.14-125 AddLeftClosedMonoidalPostComposeMorphism

`‣ AddLeftClosedMonoidalPostComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalPostComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{LeftClosedMonoidalPostComposeMorphism}(a, b, c).

##### 1.14-126 AddLeftClosedMonoidalPostComposeMorphismWithGivenObjects

`‣ AddLeftClosedMonoidalPostComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalPostComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{LeftClosedMonoidalPostComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-127 AddLeftClosedMonoidalPreComposeMorphism

`‣ AddLeftClosedMonoidalPreComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalPreComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{LeftClosedMonoidalPreComposeMorphism}(a, b, c).

##### 1.14-128 AddLeftClosedMonoidalPreComposeMorphismWithGivenObjects

`‣ AddLeftClosedMonoidalPreComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftClosedMonoidalPreComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{LeftClosedMonoidalPreComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-129 AddLeftDualOnMorphisms

`‣ AddLeftDualOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDualOnMorphisms`. F: ( alpha ) \mapsto \mathtt{LeftDualOnMorphisms}(alpha).

##### 1.14-130 AddLeftDualOnMorphismsWithGivenLeftDuals

`‣ AddLeftDualOnMorphismsWithGivenLeftDuals`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDualOnMorphismsWithGivenLeftDuals`. F: ( s, alpha, r ) \mapsto \mathtt{LeftDualOnMorphismsWithGivenLeftDuals}(s, alpha, r).

##### 1.14-131 AddLeftDualOnObjects

`‣ AddLeftDualOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDualOnObjects`. F: ( a ) \mapsto \mathtt{LeftDualOnObjects}(a).

##### 1.14-132 AddLeftInternalHomOnMorphisms

`‣ AddLeftInternalHomOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalHomOnMorphisms`. F: ( alpha, beta ) \mapsto \mathtt{LeftInternalHomOnMorphisms}(alpha, beta).

##### 1.14-133 AddLeftInternalHomOnMorphismsWithGivenLeftInternalHoms

`‣ AddLeftInternalHomOnMorphismsWithGivenLeftInternalHoms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalHomOnMorphismsWithGivenLeftInternalHoms`. F: ( s, alpha, beta, r ) \mapsto \mathtt{LeftInternalHomOnMorphismsWithGivenLeftInternalHoms}(s, alpha, beta, r).

##### 1.14-134 AddLeftInternalHomOnObjects

`‣ AddLeftInternalHomOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalHomOnObjects`. F: ( a, b ) \mapsto \mathtt{LeftInternalHomOnObjects}(a, b).

##### 1.14-135 AddLeftInternalHomToTensorProductAdjunctMorphism

`‣ AddLeftInternalHomToTensorProductAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalHomToTensorProductAdjunctMorphism`. F: ( b, c, g ) \mapsto \mathtt{LeftInternalHomToTensorProductAdjunctMorphism}(b, c, g).

##### 1.14-136 AddLeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct

`‣ AddLeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct`. F: ( b, c, g, t ) \mapsto \mathtt{LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct}(b, c, g, t).

##### 1.14-137 AddMorphismFromTensorProductToLeftInternalHom

`‣ AddMorphismFromTensorProductToLeftInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToLeftInternalHom`. F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToLeftInternalHom}(a, b).

##### 1.14-138 AddMorphismFromTensorProductToLeftInternalHomWithGivenObjects

`‣ AddMorphismFromTensorProductToLeftInternalHomWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToLeftInternalHomWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToLeftInternalHomWithGivenObjects}(s, a, b, r).

##### 1.14-139 AddMorphismToLeftBidual

`‣ AddMorphismToLeftBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToLeftBidual`. F: ( a ) \mapsto \mathtt{MorphismToLeftBidual}(a).

##### 1.14-140 AddMorphismToLeftBidualWithGivenLeftBidual

`‣ AddMorphismToLeftBidualWithGivenLeftBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToLeftBidualWithGivenLeftBidual`. F: ( a, r ) \mapsto \mathtt{MorphismToLeftBidualWithGivenLeftBidual}(a, r).

##### 1.14-141 AddTensorProductLeftDualityCompatibilityMorphism

`‣ AddTensorProductLeftDualityCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductLeftDualityCompatibilityMorphism`. F: ( a, b ) \mapsto \mathtt{TensorProductLeftDualityCompatibilityMorphism}(a, b).

##### 1.14-142 AddTensorProductLeftDualityCompatibilityMorphismWithGivenObjects

`‣ AddTensorProductLeftDualityCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductLeftDualityCompatibilityMorphismWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{TensorProductLeftDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r).

##### 1.14-143 AddTensorProductLeftInternalHomCompatibilityMorphism

`‣ AddTensorProductLeftInternalHomCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductLeftInternalHomCompatibilityMorphism`. F: ( list ) \mapsto \mathtt{TensorProductLeftInternalHomCompatibilityMorphism}(list).

##### 1.14-144 AddTensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects

`‣ AddTensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects}(source, list, range).

##### 1.14-145 AddTensorProductToLeftInternalHomAdjunctMorphism

`‣ AddTensorProductToLeftInternalHomAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToLeftInternalHomAdjunctMorphism`. F: ( a, b, f ) \mapsto \mathtt{TensorProductToLeftInternalHomAdjunctMorphism}(a, b, f).

##### 1.14-146 AddTensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom

`‣ AddTensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom`. F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom}(a, b, f, i).

##### 1.14-147 AddUniversalPropertyOfLeftDual

`‣ AddUniversalPropertyOfLeftDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `UniversalPropertyOfLeftDual`. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfLeftDual}(t, a, alpha).

##### 1.14-148 AddIsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit

`‣ AddIsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit`. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit}(a).

##### 1.14-149 AddIsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject

`‣ AddIsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject}(a).

##### 1.14-150 AddIsomorphismFromLeftInternalCoHomToObject

`‣ AddIsomorphismFromLeftInternalCoHomToObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromLeftInternalCoHomToObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromLeftInternalCoHomToObject}(a).

##### 1.14-151 AddIsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom

`‣ AddIsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom`. F: ( a, s ) \mapsto \mathtt{IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom}(a, s).

##### 1.14-152 AddIsomorphismFromObjectToLeftInternalCoHom

`‣ AddIsomorphismFromObjectToLeftInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToLeftInternalCoHom`. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToLeftInternalCoHom}(a).

##### 1.14-153 AddIsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom

`‣ AddIsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom`. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom}(a, r).

##### 1.14-154 AddLeftCoDualOnMorphisms

`‣ AddLeftCoDualOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoDualOnMorphisms`. F: ( alpha ) \mapsto \mathtt{LeftCoDualOnMorphisms}(alpha).

##### 1.14-155 AddLeftCoDualOnMorphismsWithGivenLeftCoDuals

`‣ AddLeftCoDualOnMorphismsWithGivenLeftCoDuals`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoDualOnMorphismsWithGivenLeftCoDuals`. F: ( s, alpha, r ) \mapsto \mathtt{LeftCoDualOnMorphismsWithGivenLeftCoDuals}(s, alpha, r).

##### 1.14-156 AddLeftCoDualOnObjects

`‣ AddLeftCoDualOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoDualOnObjects`. F: ( a ) \mapsto \mathtt{LeftCoDualOnObjects}(a).

##### 1.14-157 AddLeftCoDualityTensorProductCompatibilityMorphism

`‣ AddLeftCoDualityTensorProductCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoDualityTensorProductCompatibilityMorphism`. F: ( a, b ) \mapsto \mathtt{LeftCoDualityTensorProductCompatibilityMorphism}(a, b).

##### 1.14-158 AddLeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects

`‣ AddLeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects}(s, a, b, r).

##### 1.14-159 AddLeftCoclosedMonoidalCoevaluationMorphism

`‣ AddLeftCoclosedMonoidalCoevaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalCoevaluationMorphism`. F: ( a, b ) \mapsto \mathtt{LeftCoclosedMonoidalCoevaluationMorphism}(a, b).

##### 1.14-160 AddLeftCoclosedMonoidalCoevaluationMorphismWithGivenSource

`‣ AddLeftCoclosedMonoidalCoevaluationMorphismWithGivenSource`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource`. F: ( a, b, s ) \mapsto \mathtt{LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource}(a, b, s).

##### 1.14-161 AddLeftCoclosedMonoidalEvaluationForLeftCoDual

`‣ AddLeftCoclosedMonoidalEvaluationForLeftCoDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalEvaluationForLeftCoDual`. F: ( a ) \mapsto \mathtt{LeftCoclosedMonoidalEvaluationForLeftCoDual}(a).

##### 1.14-162 AddLeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct

`‣ AddLeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct}(s, a, r).

##### 1.14-163 AddLeftCoclosedMonoidalEvaluationMorphism

`‣ AddLeftCoclosedMonoidalEvaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalEvaluationMorphism`. F: ( a, b ) \mapsto \mathtt{LeftCoclosedMonoidalEvaluationMorphism}(a, b).

##### 1.14-164 AddLeftCoclosedMonoidalEvaluationMorphismWithGivenRange

`‣ AddLeftCoclosedMonoidalEvaluationMorphismWithGivenRange`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalEvaluationMorphismWithGivenRange`. F: ( a, b, r ) \mapsto \mathtt{LeftCoclosedMonoidalEvaluationMorphismWithGivenRange}(a, b, r).

##### 1.14-165 AddLeftCoclosedMonoidalLambdaElimination

`‣ AddLeftCoclosedMonoidalLambdaElimination`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalLambdaElimination`. F: ( a, b, alpha ) \mapsto \mathtt{LeftCoclosedMonoidalLambdaElimination}(a, b, alpha).

##### 1.14-166 AddLeftCoclosedMonoidalLambdaIntroduction

`‣ AddLeftCoclosedMonoidalLambdaIntroduction`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalLambdaIntroduction`. F: ( alpha ) \mapsto \mathtt{LeftCoclosedMonoidalLambdaIntroduction}(alpha).

##### 1.14-167 AddLeftCoclosedMonoidalPostCoComposeMorphism

`‣ AddLeftCoclosedMonoidalPostCoComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalPostCoComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{LeftCoclosedMonoidalPostCoComposeMorphism}(a, b, c).

##### 1.14-168 AddLeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects

`‣ AddLeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-169 AddLeftCoclosedMonoidalPreCoComposeMorphism

`‣ AddLeftCoclosedMonoidalPreCoComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalPreCoComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{LeftCoclosedMonoidalPreCoComposeMorphism}(a, b, c).

##### 1.14-170 AddLeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects

`‣ AddLeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.14-171 AddLeftInternalCoHomOnMorphisms

`‣ AddLeftInternalCoHomOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalCoHomOnMorphisms`. F: ( alpha, beta ) \mapsto \mathtt{LeftInternalCoHomOnMorphisms}(alpha, beta).

##### 1.14-172 AddLeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms

`‣ AddLeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms`. F: ( s, alpha, beta, r ) \mapsto \mathtt{LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms}(s, alpha, beta, r).

##### 1.14-173 AddLeftInternalCoHomOnObjects

`‣ AddLeftInternalCoHomOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalCoHomOnObjects`. F: ( a, b ) \mapsto \mathtt{LeftInternalCoHomOnObjects}(a, b).

##### 1.14-174 AddLeftInternalCoHomTensorProductCompatibilityMorphism

`‣ AddLeftInternalCoHomTensorProductCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalCoHomTensorProductCompatibilityMorphism`. F: ( list ) \mapsto \mathtt{LeftInternalCoHomTensorProductCompatibilityMorphism}(list).

##### 1.14-175 AddLeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

`‣ AddLeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}(source, list, range).

##### 1.14-176 AddLeftInternalCoHomToTensorProductAdjunctMorphism

`‣ AddLeftInternalCoHomToTensorProductAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalCoHomToTensorProductAdjunctMorphism`. F: ( a, c, f ) \mapsto \mathtt{LeftInternalCoHomToTensorProductAdjunctMorphism}(a, c, f).

##### 1.14-177 AddLeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct

`‣ AddLeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct`. F: ( a, c, f, t ) \mapsto \mathtt{LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct}(a, c, f, t).

##### 1.14-178 AddMorphismFromLeftCoBidual

`‣ AddMorphismFromLeftCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromLeftCoBidual`. F: ( a ) \mapsto \mathtt{MorphismFromLeftCoBidual}(a).

##### 1.14-179 AddMorphismFromLeftCoBidualWithGivenLeftCoBidual

`‣ AddMorphismFromLeftCoBidualWithGivenLeftCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromLeftCoBidualWithGivenLeftCoBidual`. F: ( a, s ) \mapsto \mathtt{MorphismFromLeftCoBidualWithGivenLeftCoBidual}(a, s).

##### 1.14-180 AddMorphismFromLeftInternalCoHomToTensorProduct

`‣ AddMorphismFromLeftInternalCoHomToTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromLeftInternalCoHomToTensorProduct`. F: ( a, b ) \mapsto \mathtt{MorphismFromLeftInternalCoHomToTensorProduct}(a, b).

##### 1.14-181 AddMorphismFromLeftInternalCoHomToTensorProductWithGivenObjects

`‣ AddMorphismFromLeftInternalCoHomToTensorProductWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects}(s, a, b, r).

##### 1.14-182 AddTensorProductToLeftInternalCoHomAdjunctMorphism

`‣ AddTensorProductToLeftInternalCoHomAdjunctMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToLeftInternalCoHomAdjunctMorphism`. F: ( b, c, g ) \mapsto \mathtt{TensorProductToLeftInternalCoHomAdjunctMorphism}(b, c, g).

##### 1.14-183 AddTensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom

`‣ AddTensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom`. F: ( b, c, g, i ) \mapsto \mathtt{TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom}(b, c, g, i).

##### 1.14-184 AddUniversalPropertyOfLeftCoDual

`‣ AddUniversalPropertyOfLeftCoDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `UniversalPropertyOfLeftCoDual`. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfLeftCoDual}(t, a, alpha).

##### 1.14-185 AddAssociatorLeftToRight

`‣ AddAssociatorLeftToRight`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `AssociatorLeftToRight`. F: ( a, b, c ) \mapsto \mathtt{AssociatorLeftToRight}(a, b, c).

##### 1.14-186 AddAssociatorLeftToRightWithGivenTensorProducts

`‣ AddAssociatorLeftToRightWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `AssociatorLeftToRightWithGivenTensorProducts`. F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorLeftToRightWithGivenTensorProducts}(s, a, b, c, r).

##### 1.14-187 AddAssociatorRightToLeft

`‣ AddAssociatorRightToLeft`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `AssociatorRightToLeft`. F: ( a, b, c ) \mapsto \mathtt{AssociatorRightToLeft}(a, b, c).

##### 1.14-188 AddAssociatorRightToLeftWithGivenTensorProducts

`‣ AddAssociatorRightToLeftWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `AssociatorRightToLeftWithGivenTensorProducts`. F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorRightToLeftWithGivenTensorProducts}(s, a, b, c, r).

##### 1.14-189 AddLeftUnitor

`‣ AddLeftUnitor`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftUnitor`. F: ( a ) \mapsto \mathtt{LeftUnitor}(a).

##### 1.14-190 AddLeftUnitorInverse

`‣ AddLeftUnitorInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftUnitorInverse`. F: ( a ) \mapsto \mathtt{LeftUnitorInverse}(a).

##### 1.14-191 AddLeftUnitorInverseWithGivenTensorProduct

`‣ AddLeftUnitorInverseWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftUnitorInverseWithGivenTensorProduct`. F: ( a, r ) \mapsto \mathtt{LeftUnitorInverseWithGivenTensorProduct}(a, r).

##### 1.14-192 AddLeftUnitorWithGivenTensorProduct

`‣ AddLeftUnitorWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftUnitorWithGivenTensorProduct`. F: ( a, s ) \mapsto \mathtt{LeftUnitorWithGivenTensorProduct}(a, s).

##### 1.14-193 AddRightUnitor

`‣ AddRightUnitor`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightUnitor`. F: ( a ) \mapsto \mathtt{RightUnitor}(a).

##### 1.14-194 AddRightUnitorInverse

`‣ AddRightUnitorInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightUnitorInverse`. F: ( a ) \mapsto \mathtt{RightUnitorInverse}(a).

##### 1.14-195 AddRightUnitorInverseWithGivenTensorProduct

`‣ AddRightUnitorInverseWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightUnitorInverseWithGivenTensorProduct`. F: ( a, r ) \mapsto \mathtt{RightUnitorInverseWithGivenTensorProduct}(a, r).

##### 1.14-196 AddRightUnitorWithGivenTensorProduct

`‣ AddRightUnitorWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightUnitorWithGivenTensorProduct`. F: ( a, s ) \mapsto \mathtt{RightUnitorWithGivenTensorProduct}(a, s).

##### 1.14-197 AddTensorProductOnMorphisms

`‣ AddTensorProductOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductOnMorphisms`. F: ( alpha, beta ) \mapsto \mathtt{TensorProductOnMorphisms}(alpha, beta).

##### 1.14-198 AddTensorProductOnMorphismsWithGivenTensorProducts

`‣ AddTensorProductOnMorphismsWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductOnMorphismsWithGivenTensorProducts`. F: ( s, alpha, beta, r ) \mapsto \mathtt{TensorProductOnMorphismsWithGivenTensorProducts}(s, alpha, beta, r).

##### 1.14-199 AddCoevaluationForDual

`‣ AddCoevaluationForDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoevaluationForDual`. F: ( a ) \mapsto \mathtt{CoevaluationForDual}(a).

##### 1.14-200 AddCoevaluationForDualWithGivenTensorProduct

`‣ AddCoevaluationForDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoevaluationForDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{CoevaluationForDualWithGivenTensorProduct}(s, a, r).

##### 1.14-201 AddIsomorphismFromInternalHomToTensorProductWithDualObject

`‣ AddIsomorphismFromInternalHomToTensorProductWithDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalHomToTensorProductWithDualObject`. F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalHomToTensorProductWithDualObject}(a, b).

##### 1.14-202 AddIsomorphismFromTensorProductWithDualObjectToInternalHom

`‣ AddIsomorphismFromTensorProductWithDualObjectToInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromTensorProductWithDualObjectToInternalHom`. F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithDualObjectToInternalHom}(a, b).

##### 1.14-203 AddMorphismFromBidual

`‣ AddMorphismFromBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromBidual`. F: ( a ) \mapsto \mathtt{MorphismFromBidual}(a).

##### 1.14-204 AddMorphismFromBidualWithGivenBidual

`‣ AddMorphismFromBidualWithGivenBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromBidualWithGivenBidual`. F: ( a, s ) \mapsto \mathtt{MorphismFromBidualWithGivenBidual}(a, s).

##### 1.14-205 AddMorphismFromInternalHomToTensorProduct

`‣ AddMorphismFromInternalHomToTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromInternalHomToTensorProduct`. F: ( a, b ) \mapsto \mathtt{MorphismFromInternalHomToTensorProduct}(a, b).

##### 1.14-206 AddMorphismFromInternalHomToTensorProductWithGivenObjects

`‣ AddMorphismFromInternalHomToTensorProductWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromInternalHomToTensorProductWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalHomToTensorProductWithGivenObjects}(s, a, b, r).

##### 1.14-207 AddRankMorphism

`‣ AddRankMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RankMorphism`. F: ( a ) \mapsto \mathtt{RankMorphism}(a).

##### 1.14-208 AddTensorProductInternalHomCompatibilityMorphismInverse

`‣ AddTensorProductInternalHomCompatibilityMorphismInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductInternalHomCompatibilityMorphismInverse`. F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverse}(list).

##### 1.14-209 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects

`‣ AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}(source, list, range).

##### 1.14-210 AddTraceMap

`‣ AddTraceMap`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TraceMap`. F: ( alpha ) \mapsto \mathtt{TraceMap}(alpha).

##### 1.14-211 AddCoRankMorphism

`‣ AddCoRankMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoRankMorphism`. F: ( a ) \mapsto \mathtt{CoRankMorphism}(a).

##### 1.14-212 AddCoTraceMap

`‣ AddCoTraceMap`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoTraceMap`. F: ( alpha ) \mapsto \mathtt{CoTraceMap}(alpha).

##### 1.14-213 AddCoclosedCoevaluationForCoDual

`‣ AddCoclosedCoevaluationForCoDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedCoevaluationForCoDual`. F: ( a ) \mapsto \mathtt{CoclosedCoevaluationForCoDual}(a).

##### 1.14-214 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct

`‣ AddCoclosedCoevaluationForCoDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedCoevaluationForCoDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{CoclosedCoevaluationForCoDualWithGivenTensorProduct}(s, a, r).

##### 1.14-215 AddInternalCoHomTensorProductCompatibilityMorphismInverse

`‣ AddInternalCoHomTensorProductCompatibilityMorphismInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomTensorProductCompatibilityMorphismInverse`. F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverse}(list).

##### 1.14-216 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects

`‣ AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}(source, list, range).

##### 1.14-217 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject

`‣ AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalCoHomToTensorProductWithCoDualObject`. F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}(a, b).

##### 1.14-218 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom

`‣ AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom`. F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}(a, b).

##### 1.14-219 AddMorphismFromTensorProductToInternalCoHom

`‣ AddMorphismFromTensorProductToInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToInternalCoHom`. F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHom}(a, b).

##### 1.14-220 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects

`‣ AddMorphismFromTensorProductToInternalCoHomWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToInternalCoHomWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHomWithGivenObjects}(s, a, b, r).

##### 1.14-221 AddMorphismToCoBidual

`‣ AddMorphismToCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToCoBidual`. F: ( a ) \mapsto \mathtt{MorphismToCoBidual}(a).

##### 1.14-222 AddMorphismToCoBidualWithGivenCoBidual

`‣ AddMorphismToCoBidualWithGivenCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToCoBidualWithGivenCoBidual`. F: ( a, r ) \mapsto \mathtt{MorphismToCoBidualWithGivenCoBidual}(a, r).

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 Ind

generated by GAPDoc2HTML