GAP (MonoidalCategories) - Chapter 1: Monoidal Categories

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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 Closed Monoidal Categories
1.5-1 InternalHomOnObjects
1.5-2 InternalHomOnMorphisms
1.5-3 InternalHomOnMorphismsWithGivenInternalHoms
1.5-4 EvaluationMorphism
1.5-5 EvaluationMorphismWithGivenSource
1.5-6 CoevaluationMorphism
1.5-7 CoevaluationMorphismWithGivenRange
1.5-8 TensorProductToInternalHomAdjunctionMap
1.5-9 InternalHomToTensorProductAdjunctionMap
1.5-10 MonoidalPreComposeMorphism
1.5-11 MonoidalPreComposeMorphismWithGivenObjects
1.5-12 MonoidalPostComposeMorphism
1.5-13 MonoidalPostComposeMorphismWithGivenObjects
1.5-14 DualOnObjects
1.5-15 DualOnMorphisms
1.5-16 DualOnMorphismsWithGivenDuals
1.5-17 EvaluationForDual
1.5-18 EvaluationForDualWithGivenTensorProduct
1.5-19 MorphismToBidual
1.5-20 MorphismToBidualWithGivenBidual
1.5-21 TensorProductInternalHomCompatibilityMorphism
1.5-22 TensorProductInternalHomCompatibilityMorphismWithGivenObjects
1.5-23 TensorProductDualityCompatibilityMorphism
1.5-24 TensorProductDualityCompatibilityMorphismWithGivenObjects
1.5-25 MorphismFromTensorProductToInternalHom
1.5-26 MorphismFromTensorProductToInternalHomWithGivenObjects
1.5-27 IsomorphismFromDualObjectToInternalHomIntoTensorUnit
1.5-28 IsomorphismFromInternalHomIntoTensorUnitToDualObject
1.5-29 UniversalPropertyOfDual
1.5-30 LambdaIntroduction
1.5-31 LambdaElimination
1.5-32 IsomorphismFromObjectToInternalHom
1.5-33 IsomorphismFromObjectToInternalHomWithGivenInternalHom
1.5-34 IsomorphismFromInternalHomToObject
1.5-35 IsomorphismFromInternalHomToObjectWithGivenInternalHom
1.6 Coclosed Monoidal Categories
1.6-1 InternalCoHomOnObjects
1.6-2 InternalCoHomOnMorphisms
1.6-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms
1.6-4 CoclosedEvaluationMorphism
1.6-5 CoclosedEvaluationMorphismWithGivenRange
1.6-6 CoclosedCoevaluationMorphism
1.6-7 CoclosedCoevaluationMorphismWithGivenSource
1.6-8 TensorProductToInternalCoHomAdjunctionMap
1.6-9 InternalCoHomToTensorProductAdjunctionMap
1.6-10 MonoidalPreCoComposeMorphism
1.6-11 MonoidalPreCoComposeMorphismWithGivenObjects
1.6-12 MonoidalPostCoComposeMorphism
1.6-13 MonoidalPostCoComposeMorphismWithGivenObjects
1.6-14 CoDualOnObjects
1.6-15 CoDualOnMorphisms
1.6-16 CoDualOnMorphismsWithGivenCoDuals
1.6-17 CoclosedEvaluationForCoDual
1.6-18 CoclosedEvaluationForCoDualWithGivenTensorProduct
1.6-19 MorphismFromCoBidual
1.6-20 MorphismFromCoBidualWithGivenCoBidual
1.6-21 InternalCoHomTensorProductCompatibilityMorphism
1.6-22 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
1.6-23 CoDualityTensorProductCompatibilityMorphism
1.6-24 CoDualityTensorProductCompatibilityMorphismWithGivenObjects
1.6-25 MorphismFromInternalCoHomToTensorProduct
1.6-26 MorphismFromInternalCoHomToTensorProductWithGivenObjects
1.6-27 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
1.6-28 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
1.6-29 UniversalPropertyOfCoDual
1.6-30 CoLambdaIntroduction
1.6-31 CoLambdaElimination
1.6-32 IsomorphismFromObjectToInternalCoHom
1.6-33 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
1.6-34 IsomorphismFromInternalCoHomToObject
1.6-35 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
1.7 Symmetric Closed Monoidal Categories
1.8 Symmetric Coclosed Monoidal Categories
1.9 Rigid Symmetric Closed Monoidal Categories
1.9-1 IsomorphismFromTensorProductWithDualObjectToInternalHom
1.9-2 IsomorphismFromInternalHomToTensorProductWithDualObject
1.9-3 MorphismFromInternalHomToTensorProduct
1.9-4 MorphismFromInternalHomToTensorProductWithGivenObjects
1.9-5 TensorProductInternalHomCompatibilityMorphismInverse
1.9-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
1.9-7 CoevaluationForDual
1.9-8 CoevaluationForDualWithGivenTensorProduct
1.9-9 TraceMap
1.9-10 RankMorphism
1.9-11 MorphismFromBidual
1.9-12 MorphismFromBidualWithGivenBidual
1.10 Rigid Symmetric Coclosed Monoidal Categories
1.10-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject
1.10-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
1.10-3 MorphismFromTensorProductToInternalCoHom
1.10-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects
1.10-5 InternalCoHomTensorProductCompatibilityMorphismInverse
1.10-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
1.10-7 CoclosedCoevaluationForCoDual
1.10-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct
1.10-9 CoTraceMap
1.10-10 CoRankMorphism
1.10-11 MorphismToCoBidual
1.10-12 MorphismToCoBidualWithGivenCoBidual
1.11 Convenience Methods
1.11-1 InternalHom
1.11-2 InternalCoHom
1.12 Add-methods
1.12-1 AddLeftDistributivityExpanding
1.12-2 AddLeftDistributivityExpandingWithGivenObjects
1.12-3 AddLeftDistributivityFactoring
1.12-4 AddLeftDistributivityFactoringWithGivenObjects
1.12-5 AddRightDistributivityExpanding
1.12-6 AddRightDistributivityExpandingWithGivenObjects
1.12-7 AddRightDistributivityFactoring
1.12-8 AddRightDistributivityFactoringWithGivenObjects
1.12-9 AddBraiding
1.12-10 AddBraidingInverse
1.12-11 AddBraidingInverseWithGivenTensorProducts
1.12-12 AddBraidingWithGivenTensorProducts
1.12-13 AddCoevaluationMorphism
1.12-14 AddCoevaluationMorphismWithGivenRange
1.12-15 AddDualOnMorphisms
1.12-16 AddDualOnMorphismsWithGivenDuals
1.12-17 AddDualOnObjects
1.12-18 AddEvaluationForDual
1.12-19 AddEvaluationForDualWithGivenTensorProduct
1.12-20 AddEvaluationMorphism
1.12-21 AddEvaluationMorphismWithGivenSource
1.12-22 AddInternalHomOnMorphisms
1.12-23 AddInternalHomOnMorphismsWithGivenInternalHoms
1.12-24 AddInternalHomOnObjects
1.12-25 AddInternalHomToTensorProductAdjunctionMap
1.12-26 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit
1.12-27 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject
1.12-28 AddIsomorphismFromInternalHomToObject
1.12-29 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom
1.12-30 AddIsomorphismFromObjectToInternalHom
1.12-31 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom
1.12-32 AddLambdaElimination
1.12-33 AddLambdaIntroduction
1.12-34 AddMonoidalPostComposeMorphism
1.12-35 AddMonoidalPostComposeMorphismWithGivenObjects
1.12-36 AddMonoidalPreComposeMorphism
1.12-37 AddMonoidalPreComposeMorphismWithGivenObjects
1.12-38 AddMorphismFromTensorProductToInternalHom
1.12-39 AddMorphismFromTensorProductToInternalHomWithGivenObjects
1.12-40 AddMorphismToBidual
1.12-41 AddMorphismToBidualWithGivenBidual
1.12-42 AddTensorProductDualityCompatibilityMorphism
1.12-43 AddTensorProductDualityCompatibilityMorphismWithGivenObjects
1.12-44 AddTensorProductInternalHomCompatibilityMorphism
1.12-45 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects
1.12-46 AddTensorProductToInternalHomAdjunctionMap
1.12-47 AddUniversalPropertyOfDual
1.12-48 AddCoDualOnMorphisms
1.12-49 AddCoDualOnMorphismsWithGivenCoDuals
1.12-50 AddCoDualOnObjects
1.12-51 AddCoDualityTensorProductCompatibilityMorphism
1.12-52 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects
1.12-53 AddCoLambdaElimination
1.12-54 AddCoLambdaIntroduction
1.12-55 AddCoclosedCoevaluationMorphism
1.12-56 AddCoclosedCoevaluationMorphismWithGivenSource
1.12-57 AddCoclosedEvaluationForCoDual
1.12-58 AddCoclosedEvaluationForCoDualWithGivenTensorProduct
1.12-59 AddCoclosedEvaluationMorphism
1.12-60 AddCoclosedEvaluationMorphismWithGivenRange
1.12-61 AddInternalCoHomOnMorphisms
1.12-62 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms
1.12-63 AddInternalCoHomOnObjects
1.12-64 AddInternalCoHomTensorProductCompatibilityMorphism
1.12-65 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
1.12-66 AddInternalCoHomToTensorProductAdjunctionMap
1.12-67 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
1.12-68 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
1.12-69 AddIsomorphismFromInternalCoHomToObject
1.12-70 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
1.12-71 AddIsomorphismFromObjectToInternalCoHom
1.12-72 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
1.12-73 AddMonoidalPostCoComposeMorphism
1.12-74 AddMonoidalPostCoComposeMorphismWithGivenObjects
1.12-75 AddMonoidalPreCoComposeMorphism
1.12-76 AddMonoidalPreCoComposeMorphismWithGivenObjects
1.12-77 AddMorphismFromCoBidual
1.12-78 AddMorphismFromCoBidualWithGivenCoBidual
1.12-79 AddMorphismFromInternalCoHomToTensorProduct
1.12-80 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects
1.12-81 AddTensorProductToInternalCoHomAdjunctionMap
1.12-82 AddUniversalPropertyOfCoDual
1.12-83 AddAssociatorLeftToRight
1.12-84 AddAssociatorLeftToRightWithGivenTensorProducts
1.12-85 AddAssociatorRightToLeft
1.12-86 AddAssociatorRightToLeftWithGivenTensorProducts
1.12-87 AddLeftUnitor
1.12-88 AddLeftUnitorInverse
1.12-89 AddLeftUnitorInverseWithGivenTensorProduct
1.12-90 AddLeftUnitorWithGivenTensorProduct
1.12-91 AddRightUnitor
1.12-92 AddRightUnitorInverse
1.12-93 AddRightUnitorInverseWithGivenTensorProduct
1.12-94 AddRightUnitorWithGivenTensorProduct
1.12-95 AddTensorProductOnMorphisms
1.12-96 AddTensorProductOnMorphismsWithGivenTensorProducts
1.12-97 AddCoevaluationForDual
1.12-98 AddCoevaluationForDualWithGivenTensorProduct
1.12-99 AddIsomorphismFromInternalHomToTensorProductWithDualObject
1.12-100 AddIsomorphismFromTensorProductWithDualObjectToInternalHom
1.12-101 AddMorphismFromBidual
1.12-102 AddMorphismFromBidualWithGivenBidual
1.12-103 AddMorphismFromInternalHomToTensorProduct
1.12-104 AddMorphismFromInternalHomToTensorProductWithGivenObjects
1.12-105 AddRankMorphism
1.12-106 AddTensorProductInternalHomCompatibilityMorphismInverse
1.12-107 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
1.12-108 AddTraceMap
1.12-109 AddCoRankMorphism
1.12-110 AddCoTraceMap
1.12-111 AddCoclosedCoevaluationForCoDual
1.12-112 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct
1.12-113 AddInternalCoHomTensorProductCompatibilityMorphismInverse
1.12-114 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
1.12-115 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject
1.12-116 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
1.12-117 AddMorphismFromTensorProductToInternalCoHom
1.12-118 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects
1.12-119 AddMorphismToCoBidual
1.12-120 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 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 \(\mathrm{\underline{Hom}}(b,-)\)) is called a closed monoidal category.

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

The corresponding GAP property is called `IsClosedMonoidalCategory`.

##### 1.5-1 InternalHomOnObjects

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

Returns: an object

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

##### 1.5-2 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 \(\mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b')\).

##### 1.5-3 InternalHomOnMorphismsWithGivenInternalHoms

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

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

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 \(\mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b')\).

##### 1.5-4 EvaluationMorphism

`‣ EvaluationMorphism`( 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 evaluation morphism \(\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b\), i.e., the counit of the tensor hom adjunction.

##### 1.5-5 EvaluationMorphismWithGivenSource

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

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

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

##### 1.5-6 CoevaluationMorphism

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

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

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

##### 1.5-7 CoevaluationMorphismWithGivenRange

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

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

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

##### 1.5-8 TensorProductToInternalHomAdjunctionMap

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

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

The arguments are 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.

##### 1.5-9 InternalHomToTensorProductAdjunctionMap

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

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

The arguments are 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.

##### 1.5-10 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 \(\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c)\).

##### 1.5-11 MonoidalPreComposeMorphismWithGivenObjects

`‣ MonoidalPreComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( 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 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 \(\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c)\).

##### 1.5-12 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 \(\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c)\).

##### 1.5-13 MonoidalPostComposeMorphismWithGivenObjects

`‣ MonoidalPostComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( 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 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 \(\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c)\).

##### 1.5-14 DualOnObjects

`‣ DualOnObjects`( a ) | ( attribute )

Returns: an object

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

##### 1.5-15 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 \(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\).

##### 1.5-16 DualOnMorphismsWithGivenDuals

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

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

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 \(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\).

##### 1.5-17 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 \(\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1\).

##### 1.5-18 EvaluationForDualWithGivenTensorProduct

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

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

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

##### 1.5-19 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 \(a \rightarrow (a^{\vee})^{\vee}\).

##### 1.5-20 MorphismToBidualWithGivenBidual

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

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

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

##### 1.5-21 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 \(\mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')\).

##### 1.5-22 TensorProductInternalHomCompatibilityMorphismWithGivenObjects

`‣ TensorProductInternalHomCompatibilityMorphismWithGivenObjects`( s, list, r ) | ( 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 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 \(\mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')\).

##### 1.5-23 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 \(\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}\).

##### 1.5-24 TensorProductDualityCompatibilityMorphismWithGivenObjects

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

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

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 \(\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}\).

##### 1.5-25 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 \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).

##### 1.5-26 MorphismFromTensorProductToInternalHomWithGivenObjects

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

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

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 \(\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).

##### 1.5-27 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 \(\mathrm{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}_{a}: a^{\vee} \rightarrow \mathrm{\underline{Hom}}(a,1)\).

##### 1.5-28 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 \(\mathrm{IsomorphismFromInternalHomIntoTensorUnitToDualObject}_{a}: \mathrm{\underline{Hom}}(a,1) \rightarrow a^{\vee}\).

##### 1.5-29 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 \(a^{\vee}\).

##### 1.5-30 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.

##### 1.5-31 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.

##### 1.5-32 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 \(a \rightarrow \mathrm{\underline{Hom}}(1,a)\).

##### 1.5-33 IsomorphismFromObjectToInternalHomWithGivenInternalHom

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

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

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

##### 1.5-34 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 \(\mathrm{\underline{Hom}}(1,a) \rightarrow a\).

##### 1.5-35 IsomorphismFromInternalHomToObjectWithGivenInternalHom

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

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

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

#### 1.6 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`.

##### 1.6-1 InternalCoHomOnObjects

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

Returns: an object

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

##### 1.6-2 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 \(\mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b)\).

##### 1.6-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms

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

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

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 \(\mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b)\).

##### 1.6-4 CoclosedEvaluationMorphism

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

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

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

##### 1.6-5 CoclosedEvaluationMorphismWithGivenRange

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

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

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

##### 1.6-6 CoclosedCoevaluationMorphism

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

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

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

##### 1.6-7 CoclosedCoevaluationMorphismWithGivenSource

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

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

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

##### 1.6-8 TensorProductToInternalCoHomAdjunctionMap

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

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

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

##### 1.6-9 InternalCoHomToTensorProductAdjunctionMap

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

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

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

##### 1.6-10 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 \(\mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b)\).

##### 1.6-11 MonoidalPreCoComposeMorphismWithGivenObjects

`‣ MonoidalPreCoComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( 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 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 \(\mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b)\).

##### 1.6-12 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 \(\mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c)\).

##### 1.6-13 MonoidalPostCoComposeMorphismWithGivenObjects

`‣ MonoidalPostCoComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( 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 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 \(\mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c)\).

##### 1.6-14 CoDualOnObjects

`‣ CoDualOnObjects`( a ) | ( attribute )

Returns: an object

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

##### 1.6-15 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 \(\alpha_{\vee}: b_{\vee} \rightarrow a_{\vee}\).

##### 1.6-16 CoDualOnMorphismsWithGivenCoDuals

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

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

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 \(\alpha_{\vee}: b^{\vee} \rightarrow a^{\vee}\).

##### 1.6-17 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 \(\mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a\).

##### 1.6-18 CoclosedEvaluationForCoDualWithGivenTensorProduct

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

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

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 \(\mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a\).

##### 1.6-19 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 \((a_{\vee})_{\vee} \rightarrow a\).

##### 1.6-20 MorphismFromCoBidualWithGivenCoBidual

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

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

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

##### 1.6-21 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 \(\mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')\).

##### 1.6-22 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

`‣ InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`( s, list, r ) | ( 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 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 \(\mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')\).

##### 1.6-23 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 \(\mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}\).

##### 1.6-24 CoDualityTensorProductCompatibilityMorphismWithGivenObjects

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

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

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 \(\mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}_{a,b}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}\).

##### 1.6-25 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 \(\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a\).

##### 1.6-26 MorphismFromInternalCoHomToTensorProductWithGivenObjects

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

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

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 \(\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow a \otimes b_{\vee}\).

##### 1.6-27 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 \(\mathrm{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}_{a}: a_{\vee} \rightarrow \mathrm{\underline{coHom}}(1,a)\).

##### 1.6-28 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 \(\mathrm{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}_{a}: \mathrm{\underline{coHom}}(1,a) \rightarrow a_{\vee}\).

##### 1.6-29 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 \(a_{\vee}\).

##### 1.6-30 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.

##### 1.6-31 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.

##### 1.6-32 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 \(a \rightarrow \mathrm{\underline{coHom}}(a,1)\).

##### 1.6-33 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

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

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

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

##### 1.6-34 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 \(\mathrm{\underline{coHom}}(a,1) \rightarrow a\).

##### 1.6-35 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom

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

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

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

#### 1.7 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`.

#### 1.8 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`.

#### 1.9 Rigid Symmetric Closed Monoidal Categories

A symmetric closed monoidal category \(\mathbf{C}\) satisfying

- the natural morphism

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

- the natural morphism

\(a \rightarrow \mathrm{\underline{Hom}}(\mathrm{\underline{Hom}}(a, 1), 1)\) 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 \(\mathrm{\underline{Hom}}(a,b) \coloneqq a^\vee \otimes b\) and, in particular, \(\mathrm{\underline{Hom}}(a,1) \coloneqq a^\vee \otimes 1\).

The corresponding GAP property is given by `IsRigidSymmetricClosedMonoidalCategory`.

##### 1.9-1 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 \(\mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)\).

##### 1.9-2 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 \(\mathrm{IsomorphismFromInternalHomToTensorProductWithDualObject}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).

##### 1.9-3 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 \(\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).

##### 1.9-4 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 \(\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b\).

##### 1.9-5 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 \(\mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')\).

##### 1.9-6 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 \(\mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')\).

##### 1.9-7 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 \(\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}\).

##### 1.9-8 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 \(\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}\).

##### 1.9-9 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 \(\mathrm{trace}_{\alpha}: 1 \rightarrow 1\).

##### 1.9-10 RankMorphism

`‣ RankMorphism`( a ) | ( attribute )

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

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

##### 1.9-11 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 \((a^{\vee})^{\vee} \rightarrow a\).

##### 1.9-12 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 \((a^{\vee})^{\vee} \rightarrow a\).

#### 1.10 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`.

##### 1.10-1 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 \(\mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObjectWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a\).

##### 1.10-2 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 \(\mathrm{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)\).

##### 1.10-3 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 \(\mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)\).

##### 1.10-4 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 \(\mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)\).

##### 1.10-5 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 \(\mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b')\).

##### 1.10-6 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 \(\mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b')\).

##### 1.10-7 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 \(\mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1\).

##### 1.10-8 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 \(\mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1\).

##### 1.10-9 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 \(\mathrm{cotrace}_{\alpha}: 1 \rightarrow 1\).

##### 1.10-10 CoRankMorphism

`‣ CoRankMorphism`( a ) | ( attribute )

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

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

##### 1.10-11 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 \(a \rightarrow (a_{\vee})_{\vee}\).

##### 1.10-12 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 \(a \rightarrow (a_{\vee})_{\vee}\).

#### 1.11 Convenience Methods

##### 1.11-1 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.

##### 1.11-2 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.

#### 1.12 Add-methods

##### 1.12-1 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`. \(F: ( a, L ) \mapsto \mathtt{LeftDistributivityExpanding}(a, L)\).

##### 1.12-2 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`. \(F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityExpandingWithGivenObjects}(s, a, L, r)\).

##### 1.12-3 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`. \(F: ( a, L ) \mapsto \mathtt{LeftDistributivityFactoring}(a, L)\).

##### 1.12-4 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`. \(F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityFactoringWithGivenObjects}(s, a, L, r)\).

##### 1.12-5 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`. \(F: ( L, a ) \mapsto \mathtt{RightDistributivityExpanding}(L, a)\).

##### 1.12-6 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`. \(F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityExpandingWithGivenObjects}(s, L, a, r)\).

##### 1.12-7 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`. \(F: ( L, a ) \mapsto \mathtt{RightDistributivityFactoring}(L, a)\).

##### 1.12-8 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`. \(F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityFactoringWithGivenObjects}(s, L, a, r)\).

##### 1.12-9 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`. \(F: ( a, b ) \mapsto \mathtt{Braiding}(a, b)\).

##### 1.12-10 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`. \(F: ( a, b ) \mapsto \mathtt{BraidingInverse}(a, b)\).

##### 1.12-11 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`. \(F: ( s, a, b, r ) \mapsto \mathtt{BraidingInverseWithGivenTensorProducts}(s, a, b, r)\).

##### 1.12-12 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`. \(F: ( s, a, b, r ) \mapsto \mathtt{BraidingWithGivenTensorProducts}(s, a, b, r)\).

##### 1.12-13 AddCoevaluationMorphism

`‣ AddCoevaluationMorphism`( 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 `CoevaluationMorphism`. \(F: ( a, b ) \mapsto \mathtt{CoevaluationMorphism}(a, b)\).

##### 1.12-14 AddCoevaluationMorphismWithGivenRange

`‣ AddCoevaluationMorphismWithGivenRange`( 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 `CoevaluationMorphismWithGivenRange`. \(F: ( a, b, r ) \mapsto \mathtt{CoevaluationMorphismWithGivenRange}(a, b, r)\).

##### 1.12-15 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`. \(F: ( alpha ) \mapsto \mathtt{DualOnMorphisms}(alpha)\).

##### 1.12-16 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`. \(F: ( s, alpha, r ) \mapsto \mathtt{DualOnMorphismsWithGivenDuals}(s, alpha, r)\).

##### 1.12-17 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`. \(F: ( a ) \mapsto \mathtt{DualOnObjects}(a)\).

##### 1.12-18 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`. \(F: ( a ) \mapsto \mathtt{EvaluationForDual}(a)\).

##### 1.12-19 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`. \(F: ( s, a, r ) \mapsto \mathtt{EvaluationForDualWithGivenTensorProduct}(s, a, r)\).

##### 1.12-20 AddEvaluationMorphism

`‣ AddEvaluationMorphism`( 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 `EvaluationMorphism`. \(F: ( a, b ) \mapsto \mathtt{EvaluationMorphism}(a, b)\).

##### 1.12-21 AddEvaluationMorphismWithGivenSource

`‣ AddEvaluationMorphismWithGivenSource`( 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 `EvaluationMorphismWithGivenSource`. \(F: ( a, b, s ) \mapsto \mathtt{EvaluationMorphismWithGivenSource}(a, b, s)\).

##### 1.12-22 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`. \(F: ( alpha, beta ) \mapsto \mathtt{InternalHomOnMorphisms}(alpha, beta)\).

##### 1.12-23 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`. \(F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalHomOnMorphismsWithGivenInternalHoms}(s, alpha, beta, r)\).

##### 1.12-24 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`. \(F: ( a, b ) \mapsto \mathtt{InternalHomOnObjects}(a, b)\).

##### 1.12-25 AddInternalHomToTensorProductAdjunctionMap

`‣ AddInternalHomToTensorProductAdjunctionMap`( 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 `InternalHomToTensorProductAdjunctionMap`. \(F: ( b, c, g ) \mapsto \mathtt{InternalHomToTensorProductAdjunctionMap}(b, c, g)\).

##### 1.12-26 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`. \(F: ( a ) \mapsto \mathtt{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}(a)\).

##### 1.12-27 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`. \(F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomIntoTensorUnitToDualObject}(a)\).

##### 1.12-28 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`. \(F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomToObject}(a)\).

##### 1.12-29 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`. \(F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalHomToObjectWithGivenInternalHom}(a, s)\).

##### 1.12-30 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`. \(F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalHom}(a)\).

##### 1.12-31 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`. \(F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalHomWithGivenInternalHom}(a, r)\).

##### 1.12-32 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`. \(F: ( a, b, alpha ) \mapsto \mathtt{LambdaElimination}(a, b, alpha)\).

##### 1.12-33 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`. \(F: ( alpha ) \mapsto \mathtt{LambdaIntroduction}(alpha)\).

##### 1.12-34 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`. \(F: ( a, b, c ) \mapsto \mathtt{MonoidalPostComposeMorphism}(a, b, c)\).

##### 1.12-35 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`. \(F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostComposeMorphismWithGivenObjects}(s, a, b, c, r)\).

##### 1.12-36 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`. \(F: ( a, b, c ) \mapsto \mathtt{MonoidalPreComposeMorphism}(a, b, c)\).

##### 1.12-37 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`. \(F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreComposeMorphismWithGivenObjects}(s, a, b, c, r)\).

##### 1.12-38 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`. \(F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalHom}(a, b)\).

##### 1.12-39 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`. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalHomWithGivenObjects}(s, a, b, r)\).

##### 1.12-40 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`. \(F: ( a ) \mapsto \mathtt{MorphismToBidual}(a)\).

##### 1.12-41 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`. \(F: ( a, r ) \mapsto \mathtt{MorphismToBidualWithGivenBidual}(a, r)\).

##### 1.12-42 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`. \(F: ( a, b ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphism}(a, b)\).

##### 1.12-43 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`. \(F: ( s, a, b, r ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r)\).

##### 1.12-44 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`. \(F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphism}(list)\).

##### 1.12-45 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`. \(F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}(source, list, range)\).

##### 1.12-46 AddTensorProductToInternalHomAdjunctionMap

`‣ AddTensorProductToInternalHomAdjunctionMap`( 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 `TensorProductToInternalHomAdjunctionMap`. \(F: ( a, b, f ) \mapsto \mathtt{TensorProductToInternalHomAdjunctionMap}(a, b, f)\).

##### 1.12-47 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`. \(F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfDual}(t, a, alpha)\).

##### 1.12-48 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`. \(F: ( alpha ) \mapsto \mathtt{CoDualOnMorphisms}(alpha)\).

##### 1.12-49 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`. \(F: ( s, alpha, r ) \mapsto \mathtt{CoDualOnMorphismsWithGivenCoDuals}(s, alpha, r)\).

##### 1.12-50 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`. \(F: ( a ) \mapsto \mathtt{CoDualOnObjects}(a)\).

##### 1.12-51 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`. \(F: ( a, b ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphism}(a, b)\).

##### 1.12-52 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`. \(F: ( s, a, b, r ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}(s, a, b, r)\).

##### 1.12-53 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`. \(F: ( a, b, alpha ) \mapsto \mathtt{CoLambdaElimination}(a, b, alpha)\).

##### 1.12-54 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`. \(F: ( alpha ) \mapsto \mathtt{CoLambdaIntroduction}(alpha)\).

##### 1.12-55 AddCoclosedCoevaluationMorphism

`‣ AddCoclosedCoevaluationMorphism`( 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 `CoclosedCoevaluationMorphism`. \(F: ( a, b ) \mapsto \mathtt{CoclosedCoevaluationMorphism}(a, b)\).

##### 1.12-56 AddCoclosedCoevaluationMorphismWithGivenSource

`‣ AddCoclosedCoevaluationMorphismWithGivenSource`( 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 `CoclosedCoevaluationMorphismWithGivenSource`. \(F: ( a, b, s ) \mapsto \mathtt{CoclosedCoevaluationMorphismWithGivenSource}(a, b, s)\).

##### 1.12-57 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`. \(F: ( a ) \mapsto \mathtt{CoclosedEvaluationForCoDual}(a)\).

##### 1.12-58 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`. \(F: ( s, a, r ) \mapsto \mathtt{CoclosedEvaluationForCoDualWithGivenTensorProduct}(s, a, r)\).

##### 1.12-59 AddCoclosedEvaluationMorphism

`‣ AddCoclosedEvaluationMorphism`( 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 `CoclosedEvaluationMorphism`. \(F: ( a, b ) \mapsto \mathtt{CoclosedEvaluationMorphism}(a, b)\).

##### 1.12-60 AddCoclosedEvaluationMorphismWithGivenRange

`‣ AddCoclosedEvaluationMorphismWithGivenRange`( 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 `CoclosedEvaluationMorphismWithGivenRange`. \(F: ( a, b, r ) \mapsto \mathtt{CoclosedEvaluationMorphismWithGivenRange}(a, b, r)\).

##### 1.12-61 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`. \(F: ( alpha, beta ) \mapsto \mathtt{InternalCoHomOnMorphisms}(alpha, beta)\).

##### 1.12-62 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`. \(F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalCoHomOnMorphismsWithGivenInternalCoHoms}(s, alpha, beta, r)\).

##### 1.12-63 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`. \(F: ( a, b ) \mapsto \mathtt{InternalCoHomOnObjects}(a, b)\).

##### 1.12-64 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`. \(F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphism}(list)\).

##### 1.12-65 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`. \(F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}(source, list, range)\).

##### 1.12-66 AddInternalCoHomToTensorProductAdjunctionMap

`‣ AddInternalCoHomToTensorProductAdjunctionMap`( 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 `InternalCoHomToTensorProductAdjunctionMap`. \(F: ( a, b, f ) \mapsto \mathtt{InternalCoHomToTensorProductAdjunctionMap}(a, b, f)\).

##### 1.12-67 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`. \(F: ( a ) \mapsto \mathtt{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}(a)\).

##### 1.12-68 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`. \(F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}(a)\).

##### 1.12-69 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`. \(F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObject}(a)\).

##### 1.12-70 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`. \(F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom}(a, s)\).

##### 1.12-71 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`. \(F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHom}(a)\).

##### 1.12-72 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`. \(F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom}(a, r)\).

##### 1.12-73 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`. \(F: ( a, b, c ) \mapsto \mathtt{MonoidalPostCoComposeMorphism}(a, b, c)\).

##### 1.12-74 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`. \(F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostCoComposeMorphismWithGivenObjects}(s, a, b, c, r)\).

##### 1.12-75 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`. \(F: ( a, b, c ) \mapsto \mathtt{MonoidalPreCoComposeMorphism}(a, b, c)\).

##### 1.12-76 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`. \(F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreCoComposeMorphismWithGivenObjects}(s, a, b, c, r)\).

##### 1.12-77 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`. \(F: ( a ) \mapsto \mathtt{MorphismFromCoBidual}(a)\).

##### 1.12-78 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`. \(F: ( a, s ) \mapsto \mathtt{MorphismFromCoBidualWithGivenCoBidual}(a, s)\).

##### 1.12-79 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`. \(F: ( a, b ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProduct}(a, b)\).

##### 1.12-80 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`. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProductWithGivenObjects}(s, a, b, r)\).

##### 1.12-81 AddTensorProductToInternalCoHomAdjunctionMap

`‣ AddTensorProductToInternalCoHomAdjunctionMap`( 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 `TensorProductToInternalCoHomAdjunctionMap`. \(F: ( c, b, g ) \mapsto \mathtt{TensorProductToInternalCoHomAdjunctionMap}(c, b, g)\).

##### 1.12-82 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`. \(F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfCoDual}(t, a, alpha)\).

##### 1.12-83 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`. \(F: ( a, b, c ) \mapsto \mathtt{AssociatorLeftToRight}(a, b, c)\).

##### 1.12-84 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`. \(F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorLeftToRightWithGivenTensorProducts}(s, a, b, c, r)\).

##### 1.12-85 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`. \(F: ( a, b, c ) \mapsto \mathtt{AssociatorRightToLeft}(a, b, c)\).

##### 1.12-86 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`. \(F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorRightToLeftWithGivenTensorProducts}(s, a, b, c, r)\).

##### 1.12-87 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`. \(F: ( a ) \mapsto \mathtt{LeftUnitor}(a)\).

##### 1.12-88 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`. \(F: ( a ) \mapsto \mathtt{LeftUnitorInverse}(a)\).

##### 1.12-89 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`. \(F: ( a, r ) \mapsto \mathtt{LeftUnitorInverseWithGivenTensorProduct}(a, r)\).

##### 1.12-90 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`. \(F: ( a, s ) \mapsto \mathtt{LeftUnitorWithGivenTensorProduct}(a, s)\).

##### 1.12-91 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`. \(F: ( a ) \mapsto \mathtt{RightUnitor}(a)\).

##### 1.12-92 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`. \(F: ( a ) \mapsto \mathtt{RightUnitorInverse}(a)\).

##### 1.12-93 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`. \(F: ( a, r ) \mapsto \mathtt{RightUnitorInverseWithGivenTensorProduct}(a, r)\).

##### 1.12-94 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`. \(F: ( a, s ) \mapsto \mathtt{RightUnitorWithGivenTensorProduct}(a, s)\).

##### 1.12-95 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`. \(F: ( alpha, beta ) \mapsto \mathtt{TensorProductOnMorphisms}(alpha, beta)\).

##### 1.12-96 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`. \(F: ( s, alpha, beta, r ) \mapsto \mathtt{TensorProductOnMorphismsWithGivenTensorProducts}(s, alpha, beta, r)\).

##### 1.12-97 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`. \(F: ( a ) \mapsto \mathtt{CoevaluationForDual}(a)\).

##### 1.12-98 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`. \(F: ( s, a, r ) \mapsto \mathtt{CoevaluationForDualWithGivenTensorProduct}(s, a, r)\).

##### 1.12-99 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`. \(F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalHomToTensorProductWithDualObject}(a, b)\).

##### 1.12-100 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`. \(F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithDualObjectToInternalHom}(a, b)\).

##### 1.12-101 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`. \(F: ( a ) \mapsto \mathtt{MorphismFromBidual}(a)\).

##### 1.12-102 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`. \(F: ( a, s ) \mapsto \mathtt{MorphismFromBidualWithGivenBidual}(a, s)\).

##### 1.12-103 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`. \(F: ( a, b ) \mapsto \mathtt{MorphismFromInternalHomToTensorProduct}(a, b)\).

##### 1.12-104 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`. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalHomToTensorProductWithGivenObjects}(s, a, b, r)\).

##### 1.12-105 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`. \(F: ( a ) \mapsto \mathtt{RankMorphism}(a)\).

##### 1.12-106 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`. \(F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverse}(list)\).

##### 1.12-107 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`. \(F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}(source, list, range)\).

##### 1.12-108 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`. \(F: ( alpha ) \mapsto \mathtt{TraceMap}(alpha)\).

##### 1.12-109 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`. \(F: ( a ) \mapsto \mathtt{CoRankMorphism}(a)\).

##### 1.12-110 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`. \(F: ( alpha ) \mapsto \mathtt{CoTraceMap}(alpha)\).

##### 1.12-111 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`. \(F: ( a ) \mapsto \mathtt{CoclosedCoevaluationForCoDual}(a)\).

##### 1.12-112 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`. \(F: ( s, a, r ) \mapsto \mathtt{CoclosedCoevaluationForCoDualWithGivenTensorProduct}(s, a, r)\).

##### 1.12-113 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`. \(F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverse}(list)\).

##### 1.12-114 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`. \(F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}(source, list, range)\).

##### 1.12-115 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`. \(F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}(a, b)\).

##### 1.12-116 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`. \(F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}(a, b)\).

##### 1.12-117 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`. \(F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHom}(a, b)\).

##### 1.12-118 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`. \(F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHomWithGivenObjects}(s, a, b, r)\).

##### 1.12-119 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`. \(F: ( a ) \mapsto \mathtt{MorphismToCoBidual}(a)\).

##### 1.12-120 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`. \(F: ( a, r ) \mapsto \mathtt{MorphismToCoBidualWithGivenCoBidual}(a, r)\).

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