  
  [1X1 [33X[0;0YMonoidal Categories[133X[101X
  
  
  [1X1.1 [33X[0;0YMonoidal Categories[133X[101X
  
  [33X[0;0YA [23X6[123X-tuple [23X( \mathbf{C}, \otimes, 1, \alpha, \lambda, \rho )[123X consisting of[133X
  
  [30X    [33X[0;6Ya category [23X\mathbf{C}[123X,[133X
  
  [30X    [33X[0;6Ya functor [23X\otimes: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}[123X
        compatible with the congruence of morphisms,[133X
  
  [30X    [33X[0;6Yan object [23X1 \in \mathbf{C}[123X,[133X
  
  [30X    [33X[0;6Ya natural isomorphism [23X\alpha_{a,b,c}: a \otimes (b \otimes c) \cong (a
        \otimes b) \otimes c[123X,[133X
  
  [30X    [33X[0;6Ya natural isomorphism [23X\lambda_{a}: 1 \otimes a \cong a[123X,[133X
  
  [30X    [33X[0;6Ya natural isomorphism [23X\rho_{a}: a \otimes 1 \cong a[123X,[133X
  
  [33X[0;0Yis called a [13Xmonoidal category[113X, if[133X
  
  [30X    [33X[0;6Yfor all objects [23Xa,b,c,d[123X, the pentagon identity holds:[133X
  
  [33X[0;0Y[23X(\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}[123X,[133X
  
  [30X    [33X[0;6Yfor all objects [23Xa,c[123X, the triangle identity holds:[133X
  
  [33X[0;0Y[23X(  \rho_a  \otimes  \mathrm{id}_c  ) \circ \alpha_{a,1,c} \sim \mathrm{id}_a
  \otimes \lambda_c[123X.[133X
  
  [33X[0;0YThe corresponding GAP property is given by [10XIsMonoidalCategory[110X.[133X
  
  [1X1.1-1 TensorProductOnMorphisms[101X
  
  [33X[1;0Y[29X[2XTensorProductOnMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a \otimes b, a' \otimes b')[123X[133X
  
  [33X[0;0YThe  arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  a',  \beta:  b
  \rightarrow b'[123X. The output is the tensor product [23X\alpha \otimes \beta[123X.[133X
  
  [1X1.1-2 TensorProductOnMorphismsWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XTensorProductOnMorphismsWithGivenTensorProducts[102X( [3Xs[103X, [3Xalpha[103X, [3Xbeta[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a \otimes b, a' \otimes b')[123X[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  =  a  \otimes b[123X, two morphisms [23X\alpha: a
  \rightarrow  a',  \beta:  b \rightarrow b'[123X, and an object [23Xr = a' \otimes b'[123X.
  The output is the tensor product [23X\alpha \otimes \beta[123X.[133X
  
  [1X1.1-3 AssociatorRightToLeft[101X
  
  [33X[1;0Y[29X[2XAssociatorRightToLeft[102X( [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b)
            \otimes c )[123X.[133X
  
  [33X[0;0YThe  arguments  are  three  objects  [23Xa,b,c[123X.  The  output  is  the associator
  [23X\alpha_{a,(b,c)}:  a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes
  c[123X.[133X
  
  [1X1.1-4 AssociatorRightToLeftWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XAssociatorRightToLeftWithGivenTensorProducts[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xc[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b)
            \otimes c )[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  = a \otimes (b \otimes c)[123X, three objects
  [23Xa,b,c[123X,  and  an  object  [23Xr  =  (a  \otimes  b)  \otimes c[123X. The output is the
  associator  [23X\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes
  b) \otimes c[123X.[133X
  
  [1X1.1-5 AssociatorLeftToRight[101X
  
  [33X[1;0Y[29X[2XAssociatorLeftToRight[102X( [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a
            \otimes (b \otimes c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  three  objects  [23Xa,b,c[123X.  The  output  is  the associator
  [23X\alpha_{(a,b),c}:  (a  \otimes b) \otimes c \rightarrow a \otimes (b \otimes
  c)[123X.[133X
  
  [1X1.1-6 AssociatorLeftToRightWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XAssociatorLeftToRightWithGivenTensorProducts[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xc[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a
            \otimes (b \otimes c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  = (a \otimes b) \otimes c[123X, three objects
  [23Xa,b,c[123X,  and  an  object  [23Xr  =  a  \otimes  (b  \otimes c)[123X. The output is the
  associator  [23X\alpha_{(a,b),c}:  (a \otimes b) \otimes c \rightarrow a \otimes
  (b \otimes c)[123X.[133X
  
  [1X1.1-7 LeftUnitor[101X
  
  [33X[1;0Y[29X[2XLeftUnitor[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(1 \otimes a, a)[123X[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X. The output is the left unitor [23X\lambda_a: 1
  \otimes a \rightarrow a[123X.[133X
  
  [1X1.1-8 LeftUnitorWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XLeftUnitorWithGivenTensorProduct[102X( [3Xa[103X, [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(1 \otimes a, a)[123X[133X
  
  [33X[0;0YThe  arguments  are an object [23Xa[123X and an object [23Xs = 1 \otimes a[123X. The output is
  the left unitor [23X\lambda_a: 1 \otimes a \rightarrow a[123X.[133X
  
  [1X1.1-9 LeftUnitorInverse[101X
  
  [33X[1;0Y[29X[2XLeftUnitorInverse[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, 1 \otimes a)[123X[133X
  
  [33X[0;0YThe  argument  is  an object [23Xa[123X. The output is the inverse of the left unitor
  [23X\lambda_a^{-1}: a \rightarrow 1 \otimes a[123X.[133X
  
  [1X1.1-10 LeftUnitorInverseWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XLeftUnitorInverseWithGivenTensorProduct[102X( [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, 1 \otimes a)[123X[133X
  
  [33X[0;0YThe argument is an object [23Xa[123X and an object [23Xr = 1 \otimes a[123X. The output is the
  inverse of the left unitor [23X\lambda_a^{-1}: a \rightarrow 1 \otimes a[123X.[133X
  
  [1X1.1-11 RightUnitor[101X
  
  [33X[1;0Y[29X[2XRightUnitor[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a \otimes 1, a)[123X[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output is the right unitor [23X\rho_a: a
  \otimes 1 \rightarrow a[123X.[133X
  
  [1X1.1-12 RightUnitorWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XRightUnitorWithGivenTensorProduct[102X( [3Xa[103X, [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a \otimes 1, a)[123X[133X
  
  [33X[0;0YThe  arguments  are an object [23Xa[123X and an object [23Xs = a \otimes 1[123X. The output is
  the right unitor [23X\rho_a: a \otimes 1 \rightarrow a[123X.[133X
  
  [1X1.1-13 RightUnitorInverse[101X
  
  [33X[1;0Y[29X[2XRightUnitorInverse[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, a \otimes 1)[123X[133X
  
  [33X[0;0YThe  argument  is an object [23Xa[123X. The output is the inverse of the right unitor
  [23X\rho_a^{-1}: a \rightarrow a \otimes 1[123X.[133X
  
  [1X1.1-14 RightUnitorInverseWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XRightUnitorInverseWithGivenTensorProduct[102X( [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, a \otimes 1)[123X[133X
  
  [33X[0;0YThe  arguments  are an object [23Xa[123X and an object [23Xr = a \otimes 1[123X. The output is
  the inverse of the right unitor [23X\rho_a^{-1}: a \rightarrow a \otimes 1[123X.[133X
  
  [1X1.1-15 TensorProductOnObjects[101X
  
  [33X[1;0Y[29X[2XTensorProductOnObjects[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,  b[123X. The output is the tensor product [23Xa
  \otimes b[123X.[133X
  
  [1X1.1-16 AddTensorProductOnObjects[101X
  
  [33X[1;0Y[29X[2XAddTensorProductOnObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductOnObjects[110X. [23XF: (a,b) \mapsto a \otimes b[123X.[133X
  
  [1X1.1-17 TensorUnit[101X
  
  [33X[1;0Y[29X[2XTensorUnit[102X( [3XC[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  argument  is  a category [23X\mathbf{C}[123X. The output is the tensor unit [23X1[123X of
  [23X\mathbf{C}[123X.[133X
  
  [1X1.1-18 AddTensorUnit[101X
  
  [33X[1;0Y[29X[2XAddTensorUnit[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function [23XF[123X to the category for the basic operation [10XTensorUnit[110X. [23XF: ( )
  \mapsto 1[123X.[133X
  
  
  [1X1.2 [33X[0;0YAdditive Monoidal Categories[133X[101X
  
  [1X1.2-1 LeftDistributivityExpanding[101X
  
  [33X[1;0Y[29X[2XLeftDistributivityExpanding[102X( [3Xa[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}(  a \otimes (b_1 \oplus \dots \oplus
            b_n), (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) )[123X[133X
  
  [33X[0;0YThe  arguments  are an object [23Xa[123X and a list of objects [23XL = (b_1, \dots, b_n)[123X.
  The  output  is the left distributivity morphism [23Xa \otimes (b_1 \oplus \dots
  \oplus b_n) \rightarrow (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)[123X.[133X
  
  [1X1.2-2 LeftDistributivityExpandingWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XLeftDistributivityExpandingWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3XL[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( s, r )[123X[133X
  
  [33X[0;0YThe  arguments are an object [23Xs = a \otimes (b_1 \oplus \dots \oplus b_n)[123X, an
  object  [23Xa[123X,  a  list  of  objects [23XL = (b_1, \dots, b_n)[123X, and an object [23Xr = (a
  \otimes  b_1)  \oplus  \dots  \oplus (a \otimes b_n)[123X. The output is the left
  distributivity morphism [23Xs \rightarrow r[123X.[133X
  
  [1X1.2-3 LeftDistributivityFactoring[101X
  
  [33X[1;0Y[29X[2XLeftDistributivityFactoring[102X( [3Xa[103X, [3XL[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( (a \otimes b_1) \oplus \dots \oplus (a
            \otimes b_n), a \otimes (b_1 \oplus \dots \oplus b_n) )[123X[133X
  
  [33X[0;0YThe  arguments  are an object [23Xa[123X and a list of objects [23XL = (b_1, \dots, b_n)[123X.
  The  output is the left distributivity morphism [23X(a \otimes b_1) \oplus \dots
  \oplus (a \otimes b_n) \rightarrow a \otimes (b_1 \oplus \dots \oplus b_n)[123X.[133X
  
  [1X1.2-4 LeftDistributivityFactoringWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XLeftDistributivityFactoringWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3XL[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( s, r )[123X[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  = (a \otimes b_1) \oplus \dots \oplus (a
  \otimes  b_n)[123X,  an object [23Xa[123X, a list of objects [23XL = (b_1, \dots, b_n)[123X, and an
  object  [23Xr  = a \otimes (b_1 \oplus \dots \oplus b_n)[123X. The output is the left
  distributivity morphism [23Xs \rightarrow r[123X.[133X
  
  [1X1.2-5 RightDistributivityExpanding[101X
  
  [33X[1;0Y[29X[2XRightDistributivityExpanding[102X( [3XL[103X, [3Xa[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( (b_1 \oplus \dots \oplus b_n) \otimes
            a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) )[123X[133X
  
  [33X[0;0YThe  arguments  are a list of objects [23XL = (b_1, \dots, b_n)[123X and an object [23Xa[123X.
  The  output  is  the  right distributivity morphism [23X(b_1 \oplus \dots \oplus
  b_n)  \otimes a \rightarrow (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes
  a)[123X.[133X
  
  [1X1.2-6 RightDistributivityExpandingWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XRightDistributivityExpandingWithGivenObjects[102X( [3Xs[103X, [3XL[103X, [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( s, r )[123X[133X
  
  [33X[0;0YThe  arguments  are an object [23Xs = (b_1 \oplus \dots \oplus b_n) \otimes a[123X, a
  list  of  objects [23XL = (b_1, \dots, b_n)[123X, an object [23Xa[123X, and an object [23Xr = (b_1
  \otimes  a)  \oplus  \dots  \oplus  (b_n \otimes a)[123X. The output is the right
  distributivity morphism [23Xs \rightarrow r[123X.[133X
  
  [1X1.2-7 RightDistributivityFactoring[101X
  
  [33X[1;0Y[29X[2XRightDistributivityFactoring[102X( [3XL[103X, [3Xa[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}( (b_1 \otimes a) \oplus \dots \oplus
            (b_n \otimes a), (b_1 \oplus \dots \oplus b_n) \otimes a)[123X[133X
  
  [33X[0;0YThe  arguments  are a list of objects [23XL = (b_1, \dots, b_n)[123X and an object [23Xa[123X.
  The output is the right distributivity morphism [23X(b_1 \otimes a) \oplus \dots
  \oplus (b_n \otimes a) \rightarrow (b_1 \oplus \dots \oplus b_n) \otimes a [123X.[133X
  
  [1X1.2-8 RightDistributivityFactoringWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XRightDistributivityFactoringWithGivenObjects[102X( [3Xs[103X, [3XL[103X, [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( s, r )[123X[133X
  
  [33X[0;0YThe  arguments  are  an  object [23Xs = (b_1 \otimes a) \oplus \dots \oplus (b_n
  \otimes  a)[123X,  a  list  of objects [23XL = (b_1, \dots, b_n)[123X, an object [23Xa[123X, and an
  object  [23Xr = (b_1 \oplus \dots \oplus b_n) \otimes a[123X. The output is the right
  distributivity morphism [23Xs \rightarrow r[123X.[133X
  
  
  [1X1.3 [33X[0;0YBraided Monoidal Categories[133X[101X
  
  [33X[0;0YA  monoidal category [23X\mathbf{C}[123X equipped with a natural isomorphism [23XB_{a,b}:
  a \otimes b \cong b \otimes a[123X is called a [13Xbraided monoidal category[113X if[133X
  
  [30X    [33X[0;6Y[23X\lambda_a \circ B_{a,1} \sim \rho_a[123X,[133X
  
  [30X    [33X[0;6Y[23X(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}[123X,[133X
  
  [30X    [33X[0;6Y[23X(  \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}[123X.[133X
  
  [33X[0;0YThe corresponding GAP property is given by [10XIsBraidedMonoidalCategory[110X.[133X
  
  [1X1.3-1 Braiding[101X
  
  [33X[1;0Y[29X[2XBraiding[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a \otimes b, b \otimes a )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two objects [23Xa,b[123X. The output is the braiding [23X B_{a,b}: a
  \otimes b \rightarrow b \otimes a[123X.[133X
  
  [1X1.3-2 BraidingWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XBraidingWithGivenTensorProducts[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a \otimes b, b \otimes a )[123X.[133X
  
  [33X[0;0YThe  arguments are an object [23Xs = a \otimes b[123X, two objects [23Xa,b[123X, and an object
  [23Xr  =  b  \otimes  a[123X.  The  output  is  the  braiding  [23X  B_{a,b}: a \otimes b
  \rightarrow b \otimes a[123X.[133X
  
  [1X1.3-3 BraidingInverse[101X
  
  [33X[1;0Y[29X[2XBraidingInverse[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( b \otimes a, a \otimes b )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,b[123X.  The output is the inverse braiding [23X
  B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b[123X.[133X
  
  [1X1.3-4 BraidingInverseWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XBraidingInverseWithGivenTensorProducts[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( b \otimes a, a \otimes b )[123X.[133X
  
  [33X[0;0YThe  arguments are an object [23Xs = b \otimes a[123X, two objects [23Xa,b[123X, and an object
  [23Xr = a \otimes b[123X. The output is the inverse braiding [23X B_{a,b}^{-1}: b \otimes
  a \rightarrow a \otimes b[123X.[133X
  
  
  [1X1.4 [33X[0;0YSymmetric Monoidal Categories[133X[101X
  
  [33X[0;0YA braided monoidal category [23X\mathbf{C}[123X is called [13Xsymmetric monoidal category[113X
  if  [23XB_{a,b}^{-1}  \sim  B_{b,a}[123X.  The corresponding GAP property is given by
  [10XIsSymmetricMonoidalCategory[110X.[133X
  
  
  [1X1.5 [33X[0;0YClosed Monoidal Categories[133X[101X
  
  [33X[0;0YA  monoidal  category  [23X\mathbf{C}[123X  which  has  for each functor [23X- \otimes b:
  \mathbf{C}    \rightarrow   \mathbf{C}[123X   a   right   adjoint   (denoted   by
  [23X\mathrm{\underline{Hom}}(b,-)[123X) is called a [13Xclosed monoidal category[113X.[133X
  
  [33X[0;0YIf  no  operations  involving duals are installed manually, the dual objects
  will be derived as [23Xa^\vee \coloneqq \mathrm{\underline{Hom}}(a,1)[123X.[133X
  
  [33X[0;0YThe corresponding GAP property is called [10XIsClosedMonoidalCategory[110X.[133X
  
  [1X1.5-1 InternalHomOnObjects[101X
  
  [33X[1;0Y[29X[2XInternalHomOnObjects[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  arguments  are  two  objects [23Xa,b[123X. The output is the internal hom object
  [23X\mathrm{\underline{Hom}}(a,b)[123X.[133X
  
  [1X1.5-2 InternalHomOnMorphisms[101X
  
  [33X[1;0Y[29X[2XInternalHomOnMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in   [23X\mathrm{Hom}(  \mathrm{\underline{Hom}}(a',b),
            \mathrm{\underline{Hom}}(a,b') )[123X[133X
  
  [33X[0;0YThe  arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  a',  \beta:  b
  \rightarrow    b'[123X.    The    output    is    the   internal   hom   morphism
  [23X\mathrm{\underline{Hom}}(\alpha,\beta):       \mathrm{\underline{Hom}}(a',b)
  \rightarrow \mathrm{\underline{Hom}}(a,b')[123X.[133X
  
  [1X1.5-3 InternalHomOnMorphismsWithGivenInternalHoms[101X
  
  [33X[1;0Y[29X[2XInternalHomOnMorphismsWithGivenInternalHoms[102X( [3Xs[103X, [3Xalpha[103X, [3Xbeta[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in   [23X\mathrm{Hom}(  \mathrm{\underline{Hom}}(a',b),
            \mathrm{\underline{Hom}}(a,b') )[123X[133X
  
  [33X[0;0YThe   arguments  are  an  object  [23Xs  =  \mathrm{\underline{Hom}}(a',b)[123X,  two
  morphisms [23X\alpha: a \rightarrow a', \beta: b \rightarrow b'[123X, and an object [23Xr
  =  \mathrm{\underline{Hom}}(a,b')[123X.  The  output is the internal hom morphism
  [23X\mathrm{\underline{Hom}}(\alpha,\beta):       \mathrm{\underline{Hom}}(a',b)
  \rightarrow \mathrm{\underline{Hom}}(a,b')[123X.[133X
  
  [1X1.5-4 EvaluationMorphism[101X
  
  [33X[1;0Y[29X[2XEvaluationMorphism[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes
            a, b )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two objects [23Xa, b[123X. The output is the evaluation morphism
  [23X\mathrm{ev}_{a,b}:  \mathrm{\underline{Hom}}(a,b)  \otimes  a \rightarrow b[123X,
  i.e., the counit of the tensor hom adjunction.[133X
  
  [1X1.5-5 EvaluationMorphismWithGivenSource[101X
  
  [33X[1;0Y[29X[2XEvaluationMorphismWithGivenSource[102X( [3Xa[103X, [3Xb[103X, [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes
            a, b )[123X.[133X
  
  [33X[0;0YThe    arguments    are    two    objects    [23Xa,b[123X   and   an   object   [23Xs   =
  \mathrm{\underline{Hom}}(a,b)  \otimes  a[123X.  The  output  is  the  evaluation
  morphism    [23X\mathrm{ev}_{a,b}:   \mathrm{\underline{Hom}}(a,b)   \otimes   a
  \rightarrow b[123X, i.e., the counit of the tensor hom adjunction.[133X
  
  [1X1.5-6 CoevaluationMorphism[101X
  
  [33X[1;0Y[29X[2XCoevaluationMorphism[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}(  a,  \mathrm{\underline{Hom}}(b,  a
            \otimes b) )[123X.[133X
  
  [33X[0;0YThe  arguments  are two objects [23Xa,b[123X. The output is the coevaluation morphism
  [23X\mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}}(b, a \otimes b)[123X,
  i.e., the unit of the tensor hom adjunction.[133X
  
  [1X1.5-7 CoevaluationMorphismWithGivenRange[101X
  
  [33X[1;0Y[29X[2XCoevaluationMorphismWithGivenRange[102X( [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}(  a,  \mathrm{\underline{Hom}}(b,  a
            \otimes b) )[123X.[133X
  
  [33X[0;0YThe    arguments    are    two    objects    [23Xa,b[123X   and   an   object   [23Xr   =
  \mathrm{\underline{Hom}}(b,  a  \otimes  b)[123X.  The output is the coevaluation
  morphism  [23X\mathrm{coev}_{a,b}:  a  \rightarrow \mathrm{\underline{Hom}}(b, a
  \otimes b)[123X, i.e., the unit of the tensor hom adjunction.[133X
  
  [1X1.5-8 TensorProductToInternalHomAdjunctionMap[101X
  
  [33X[1;0Y[29X[2XTensorProductToInternalHomAdjunctionMap[102X( [3Xa[103X, [3Xb[103X, [3Xf[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are objects [23Xa,b[123X and a morphism [23Xf: a \otimes b \rightarrow c[123X.
  The  output  is  a  morphism  [23Xg: a \rightarrow \mathrm{\underline{Hom}}(b,c)[123X
  corresponding to [23Xf[123X under the tensor hom adjunction.[133X
  
  [1X1.5-9 InternalHomToTensorProductAdjunctionMap[101X
  
  [33X[1;0Y[29X[2XInternalHomToTensorProductAdjunctionMap[102X( [3Xb[103X, [3Xc[103X, [3Xg[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a \otimes b, c)[123X.[133X
  
  [33X[0;0YThe   arguments   are   objects   [23Xb,c[123X   and  a  morphism  [23Xg:  a  \rightarrow
  \mathrm{\underline{Hom}}(b,c)[123X.  The  output  is  a  morphism  [23Xf: a \otimes b
  \rightarrow c[123X corresponding to [23Xg[123X under the tensor hom adjunction.[133X
  
  [1X1.5-10 MonoidalPreComposeMorphism[101X
  
  [33X[1;0Y[29X[2XMonoidalPreComposeMorphism[102X( [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes
            \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  three  objects  [23Xa,b,c[123X. The output is the precomposition
  morphism        [23X\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}:
  \mathrm{\underline{Hom}}(a,b)      \otimes     \mathrm{\underline{Hom}}(b,c)
  \rightarrow \mathrm{\underline{Hom}}(a,c)[123X.[133X
  
  [1X1.5-11 MonoidalPreComposeMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XMonoidalPreComposeMorphismWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xc[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes
            \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  =  \mathrm{\underline{Hom}}(a,b) \otimes
  \mathrm{\underline{Hom}}(b,c)[123X,  three  objects  [23Xa,b,c[123X,  and  an  object  [23Xr =
  \mathrm{\underline{Hom}}(a,c)[123X.  The  output  is  the precomposition morphism
  [23X\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}:
  \mathrm{\underline{Hom}}(a,b)      \otimes     \mathrm{\underline{Hom}}(b,c)
  \rightarrow \mathrm{\underline{Hom}}(a,c)[123X.[133X
  
  [1X1.5-12 MonoidalPostComposeMorphism[101X
  
  [33X[1;0Y[29X[2XMonoidalPostComposeMorphism[102X( [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes
            \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  three  objects [23Xa,b,c[123X. The output is the postcomposition
  morphism       [23X\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}:
  \mathrm{\underline{Hom}}(b,c)      \otimes     \mathrm{\underline{Hom}}(a,b)
  \rightarrow \mathrm{\underline{Hom}}(a,c)[123X.[133X
  
  [1X1.5-13 MonoidalPostComposeMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XMonoidalPostComposeMorphismWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xc[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes
            \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  =  \mathrm{\underline{Hom}}(b,c) \otimes
  \mathrm{\underline{Hom}}(a,b)[123X,  three  objects  [23Xa,b,c[123X,  and  an  object  [23Xr =
  \mathrm{\underline{Hom}}(a,c)[123X.  The  output  is the postcomposition morphism
  [23X\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}:
  \mathrm{\underline{Hom}}(b,c)      \otimes     \mathrm{\underline{Hom}}(a,b)
  \rightarrow \mathrm{\underline{Hom}}(a,c)[123X.[133X
  
  [1X1.5-14 DualOnObjects[101X
  
  [33X[1;0Y[29X[2XDualOnObjects[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe argument is an object [23Xa[123X. The output is its dual object [23Xa^{\vee}[123X.[133X
  
  [1X1.5-15 DualOnMorphisms[101X
  
  [33X[1;0Y[29X[2XDualOnMorphisms[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( b^{\vee}, a^{\vee} )[123X.[133X
  
  [33X[0;0YThe  argument  is a morphism [23X\alpha: a \rightarrow b[123X. The output is its dual
  morphism [23X\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}[123X.[133X
  
  [1X1.5-16 DualOnMorphismsWithGivenDuals[101X
  
  [33X[1;0Y[29X[2XDualOnMorphismsWithGivenDuals[102X( [3Xs[103X, [3Xalpha[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( b^{\vee}, a^{\vee} )[123X.[133X
  
  [33X[0;0YThe  argument is an object [23Xs = b^{\vee}[123X, a morphism [23X\alpha: a \rightarrow b[123X,
  and  an  object [23Xr = a^{\vee}[123X. The output is the dual morphism [23X\alpha^{\vee}:
  b^{\vee} \rightarrow a^{\vee}[123X.[133X
  
  [1X1.5-17 EvaluationForDual[101X
  
  [33X[1;0Y[29X[2XEvaluationForDual[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a^{\vee} \otimes a, 1 )[123X.[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output  is  the  evaluation morphism
  [23X\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1[123X.[133X
  
  [1X1.5-18 EvaluationForDualWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XEvaluationForDualWithGivenTensorProduct[102X( [3Xs[103X, [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( a^{\vee} \otimes a, 1 )[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs = a^{\vee} \otimes a[123X, an object [23Xa[123X, and an
  object  [23Xr  =  1[123X.  The  output  is  the  evaluation morphism [23X\mathrm{ev}_{a}:
  a^{\vee} \otimes a \rightarrow 1[123X.[133X
  
  [1X1.5-19 MorphismToBidual[101X
  
  [33X[1;0Y[29X[2XMorphismToBidual[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, (a^{\vee})^{\vee})[123X.[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X. The output is the morphism to the bidual [23Xa
  \rightarrow (a^{\vee})^{\vee}[123X.[133X
  
  [1X1.5-20 MorphismToBidualWithGivenBidual[101X
  
  [33X[1;0Y[29X[2XMorphismToBidualWithGivenBidual[102X( [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, (a^{\vee})^{\vee})[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xa[123X, and an object [23Xr = (a^{\vee})^{\vee}[123X. The
  output is the morphism to the bidual [23Xa \rightarrow (a^{\vee})^{\vee}[123X.[133X
  
  [1X1.5-21 TensorProductInternalHomCompatibilityMorphism[101X
  
  [33X[1;0Y[29X[2XTensorProductInternalHomCompatibilityMorphism[102X( [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes
            \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes
            b,a' \otimes b'))[123X.[133X
  
  [33X[0;0YThe  argument  is a list of four objects [23X[ a, a', b, b' ][123X. The output is the
  natural                                                             morphism
  [23X\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')[123X.[133X
  
  [1X1.5-22 TensorProductInternalHomCompatibilityMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XTensorProductInternalHomCompatibilityMorphismWithGivenObjects[102X( [3Xs[103X, [3Xlist[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes
            \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes
            b,a' \otimes b'))[123X.[133X
  
  [33X[0;0YThe arguments are a list of four objects [23X[ a, a', b, b' ][123X, and two objects [23Xs
  =  \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')[123X and
  [23Xr  =  \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')[123X. The output is the
  natural                                                             morphism
  [23X\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')[123X.[133X
  
  [1X1.5-23 TensorProductDualityCompatibilityMorphism[101X
  
  [33X[1;0Y[29X[2XTensorProductDualityCompatibilityMorphism[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes
            b)^{\vee} )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,b[123X.  The  output is the natural morphism
  [23X\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}: a^{\vee}
  \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}[123X.[133X
  
  [1X1.5-24 TensorProductDualityCompatibilityMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XTensorProductDualityCompatibilityMorphismWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism in [23X\mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes
            b)^{\vee} )[123X.[133X
  
  [33X[0;0YThe  arguments are an object [23Xs = a^{\vee} \otimes b^{\vee}[123X, two objects [23Xa,b[123X,
  and  an  object [23Xr = (a \otimes b)^{\vee}[123X. The output is the natural morphism
  [23X\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}:
  a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}[123X.[133X
  
  [1X1.5-25 MorphismFromTensorProductToInternalHom[101X
  
  [33X[1;0Y[29X[2XMorphismFromTensorProductToInternalHom[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya     morphism    in    [23X\mathrm{Hom}(    a^{\vee}    \otimes    b,
            \mathrm{\underline{Hom}}(a,b) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,b[123X.  The  output is the natural morphism
  [23X\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}:
  a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)[123X.[133X
  
  [1X1.5-26 MorphismFromTensorProductToInternalHomWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XMorphismFromTensorProductToInternalHomWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya     morphism    in    [23X\mathrm{Hom}(    a^{\vee}    \otimes    b,
            \mathrm{\underline{Hom}}(a,b) )[123X.[133X
  
  [33X[0;0YThe  arguments are an object [23Xs = a^{\vee} \otimes b[123X, two objects [23Xa,b[123X, and an
  object [23Xr = \mathrm{\underline{Hom}}(a,b)[123X. The output is the natural morphism
  [23X\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}:
  a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)[123X.[133X
  
  [1X1.5-27 IsomorphismFromDualObjectToInternalHomIntoTensorUnit[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromDualObjectToInternalHomIntoTensorUnit[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya           morphism           in           [23X\mathrm{Hom}(a^{\vee},
            \mathrm{\underline{Hom}}(a,1))[123X.[133X
  
  [33X[0;0YThe   argument   is   an   object   [23Xa[123X.   The   output   is  the  isomorphism
  [23X\mathrm{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}_{a}:  a^{\vee}
  \rightarrow \mathrm{\underline{Hom}}(a,1)[123X.[133X
  
  [1X1.5-28 IsomorphismFromInternalHomIntoTensorUnitToDualObject[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromInternalHomIntoTensorUnitToDualObject[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya    morphism    in    [23X\mathrm{Hom}(\mathrm{\underline{Hom}}(a,1),
            a^{\vee})[123X.[133X
  
  [33X[0;0YThe   argument   is   an   object   [23Xa[123X.   The   output   is  the  isomorphism
  [23X\mathrm{IsomorphismFromInternalHomIntoTensorUnitToDualObject}_{a}:
  \mathrm{\underline{Hom}}(a,1) \rightarrow a^{\vee}[123X.[133X
  
  [1X1.5-29 UniversalPropertyOfDual[101X
  
  [33X[1;0Y[29X[2XUniversalPropertyOfDual[102X( [3Xt[103X, [3Xa[103X, [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(t, a^{\vee})[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xt,a[123X,  and  a morphism [23X\alpha: t \otimes a
  \rightarrow  1[123X.  The  output is the morphism [23Xt \rightarrow a^{\vee}[123X given by
  the universal property of [23Xa^{\vee}[123X.[133X
  
  [1X1.5-30 LambdaIntroduction[101X
  
  [33X[1;0Y[29X[2XLambdaIntroduction[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( 1, \mathrm{\underline{Hom}}(a,b) )[123X.[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha:  a  \rightarrow b[123X. The output is the
  corresponding morphism [23X1 \rightarrow \mathrm{\underline{Hom}}(a,b)[123X under the
  tensor hom adjunction.[133X
  
  [1X1.5-31 LambdaElimination[101X
  
  [33X[1;0Y[29X[2XLambdaElimination[102X( [3Xa[103X, [3Xb[103X, [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,b[123X, and a morphism [23X\alpha: 1 \rightarrow
  \mathrm{\underline{Hom}}(a,b)[123X.  The  output  is  a  morphism [23Xa \rightarrow b[123X
  corresponding to [23X\alpha[123X under the tensor hom adjunction.[133X
  
  [1X1.5-32 IsomorphismFromObjectToInternalHom[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromObjectToInternalHom[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a))[123X.[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output  is the natural isomorphism [23Xa
  \rightarrow \mathrm{\underline{Hom}}(1,a)[123X.[133X
  
  [1X1.5-33 IsomorphismFromObjectToInternalHomWithGivenInternalHom[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromObjectToInternalHomWithGivenInternalHom[102X( [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a))[123X.[133X
  
  [33X[0;0YThe    argument    is    an    object    [23Xa[123X,    and    an    object    [23Xr    =
  \mathrm{\underline{Hom}}(1,a)[123X.  The  output  is  the  natural  isomorphism [23Xa
  \rightarrow \mathrm{\underline{Hom}}(1,a)[123X.[133X
  
  [1X1.5-34 IsomorphismFromInternalHomToObject[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromInternalHomToObject[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a)[123X.[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output  is  the  natural isomorphism
  [23X\mathrm{\underline{Hom}}(1,a) \rightarrow a[123X.[133X
  
  [1X1.5-35 IsomorphismFromInternalHomToObjectWithGivenInternalHom[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromInternalHomToObjectWithGivenInternalHom[102X( [3Xa[103X, [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a)[123X.[133X
  
  [33X[0;0YThe    argument    is    an    object    [23Xa[123X,    and    an    object    [23Xs    =
  \mathrm{\underline{Hom}}(1,a)[123X.   The   output  is  the  natural  isomorphism
  [23X\mathrm{\underline{Hom}}(1,a) \rightarrow a[123X.[133X
  
  
  [1X1.6 [33X[0;0YCoclosed Monoidal Categories[133X[101X
  
  [33X[0;0YA  monoidal  category  [23X\mathbf{C}[123X  which  has  for each functor [23X- \otimes b:
  \mathbf{C}    \rightarrow    \mathbf{C}[123X   a   left   adjoint   (denoted   by
  [23X\mathrm{\underline{coHom}}(-,b)[123X) is called a [13Xcoclosed monoidal category[113X.[133X
  
  [33X[0;0YIf  no  operations  involving  coduals  are  installed  manually, the codual
  objects will be derived as [23Xa_\vee \coloneqq \mathrm{\underline{coHom}}(1,a)[123X.[133X
  
  [33X[0;0YThe corresponding GAP property is called [10XIsCoclosedMonoidalCategory[110X.[133X
  
  [1X1.6-1 InternalCoHomOnObjects[101X
  
  [33X[1;0Y[29X[2XInternalCoHomOnObjects[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe  arguments  are two objects [23Xa,b[123X. The output is the internal cohom object
  [23X\mathrm{\underline{coHom}}(a,b)[123X.[133X
  
  [1X1.6-2 InternalCoHomOnMorphisms[101X
  
  [33X[1;0Y[29X[2XInternalCoHomOnMorphisms[102X( [3Xalpha[103X, [3Xbeta[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism  in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,b'),
            \mathrm{\underline{coHom}}(a',b) )[123X[133X
  
  [33X[0;0YThe  arguments  are  two  morphisms  [23X\alpha:  a  \rightarrow  a',  \beta:  b
  \rightarrow    b'[123X.    The    output   is   the   internal   cohom   morphism
  [23X\mathrm{\underline{coHom}}(\alpha,\beta):   \mathrm{\underline{coHom}}(a,b')
  \rightarrow \mathrm{\underline{coHom}}(a',b)[123X.[133X
  
  [1X1.6-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms[101X
  
  [33X[1;0Y[29X[2XInternalCoHomOnMorphismsWithGivenInternalCoHoms[102X( [3Xs[103X, [3Xalpha[103X, [3Xbeta[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism  in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,b'),
            \mathrm{\underline{coHom}}(a',b) )[123X[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  =  \mathrm{\underline{coHom}}(a,b')[123X, two
  morphisms [23X\alpha: a \rightarrow a', \beta: b \rightarrow b'[123X, and an object [23Xr
  =   \mathrm{\underline{coHom}}(a',b)[123X.  The  output  is  the  internal  cohom
  morphism                           [23X\mathrm{\underline{coHom}}(\alpha,\beta):
  \mathrm{\underline{coHom}}(a,b')                                 \rightarrow
  \mathrm{\underline{coHom}}(a',b)[123X.[133X
  
  [1X1.6-4 CoclosedEvaluationMorphism[101X
  
  [33X[1;0Y[29X[2XCoclosedEvaluationMorphism[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}(  a, \mathrm{\underline{coHom}}(a,b)
            \otimes b )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two objects [23Xa, b[123X. The output is the coclosed evaluation
  morphism            [23X\mathrm{coclev}_{a,b}:           a           \rightarrow
  \mathrm{\underline{coHom}}(a,b)  \otimes  b[123X,  i.e.,  the  unit  of the cohom
  tensor adjunction.[133X
  
  [1X1.6-5 CoclosedEvaluationMorphismWithGivenRange[101X
  
  [33X[1;0Y[29X[2XCoclosedEvaluationMorphismWithGivenRange[102X( [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}(  a, \mathrm{\underline{coHom}}(a,b)
            \otimes b )[123X.[133X
  
  [33X[0;0YThe    arguments    are    two    objects    [23Xa,b[123X   and   an   object   [23Xr   =
  \mathrm{\underline{coHom}}(a,b)  \otimes  b[123X.  The  output  is  the  coclosed
  evaluation      morphism      [23X\mathrm{coclev}_{a,b}:      a      \rightarrow
  \mathrm{\underline{coHom}}(a,b)  \otimes  b[123X,  i.e.,  the  unit  of the cohom
  tensor adjunction.[133X
  
  [1X1.6-6 CoclosedCoevaluationMorphism[101X
  
  [33X[1;0Y[29X[2XCoclosedCoevaluationMorphism[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in [23X\mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes
            b, b), a )[123X.[133X
  
  [33X[0;0YThe  arguments  are two objects [23Xa,b[123X. The output is the coclosed coevaluation
  morphism [23X\mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, b)
  \rightarrow a[123X, i.e., the counit of the cohom tensor adjunction.[133X
  
  [1X1.6-7 CoclosedCoevaluationMorphismWithGivenSource[101X
  
  [33X[1;0Y[29X[2XCoclosedCoevaluationMorphismWithGivenSource[102X( [3Xa[103X, [3Xb[103X, [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in [23X\mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes
            b, b), b )[123X.[133X
  
  [33X[0;0YThe    arguments    are    two    objects    [23Xa,b[123X   and   an   object   [23Xs   =
  \mathrm{\underline{coHom}(a  \otimes  b,  b)}[123X.  The  output  is the coclosed
  coevaluation  morphism [23X\mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a
  \otimes b, b) \rightarrow a[123X, i.e., the unit of the cohom tensor adjunction.[133X
  
  [1X1.6-8 TensorProductToInternalCoHomAdjunctionMap[101X
  
  [33X[1;0Y[29X[2XTensorProductToInternalCoHomAdjunctionMap[102X( [3Xc[103X, [3Xb[103X, [3Xg[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c )[123X.[133X
  
  [33X[0;0YThe  arguments  are objects [23Xc,b[123X and a morphism [23Xg: a \rightarrow c \otimes b[123X.
  The  output  is  a morphism [23Xf: \mathrm{\underline{coHom}}(a,b) \rightarrow c[123X
  corresponding to [23Xg[123X under the cohom tensor adjunction.[133X
  
  [1X1.6-9 InternalCoHomToTensorProductAdjunctionMap[101X
  
  [33X[1;0Y[29X[2XInternalCoHomToTensorProductAdjunctionMap[102X( [3Xa[103X, [3Xb[103X, [3Xf[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, c \otimes b)[123X.[133X
  
  [33X[0;0YThe     arguments     are     objects     [23Xa,b[123X     and    a    morphism    [23Xf:
  \mathrm{\underline{coHom}}(a,b) \rightarrow c[123X. The output is a morphism [23Xg: a
  \rightarrow   c  \otimes  b[123X  corresponding  to  [23Xf[123X  under  the  cohom  tensor
  adjunction.[133X
  
  [1X1.6-10 MonoidalPreCoComposeMorphism[101X
  
  [33X[1;0Y[29X[2XMonoidalPreCoComposeMorphism[102X( [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,c),
            \mathrm{\underline{coHom}}(b,c)                            \otimes
            \mathrm{\underline{coHom}}(a,b) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  three objects [23Xa,b,c[123X. The output is the precocomposition
  morphism      [23X\mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}:
  \mathrm{\underline{coHom}}(a,c)  \rightarrow \mathrm{\underline{coHom}}(b,c)
  \otimes \mathrm{\underline{coHom}}(a,b)[123X.[133X
  
  [1X1.6-11 MonoidalPreCoComposeMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XMonoidalPreCoComposeMorphismWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xc[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,c),
            \mathrm{\underline{coHom}}(b,c)                            \otimes
            \mathrm{\underline{coHom}}(a,b) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  = \mathrm{\underline{coHom}}(a,c)[123X, three
  objects  [23Xa,b,c[123X,  and  an  object [23Xr = \mathrm{\underline{coHom}}(a,b) \otimes
  \mathrm{\underline{coHom}}(b,c)[123X. The output is the precocomposition morphism
  [23X\mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}:
  \mathrm{\underline{coHom}}(a,c)  \rightarrow \mathrm{\underline{coHom}}(b,c)
  \otimes \mathrm{\underline{coHom}}(a,b)[123X.[133X
  
  [1X1.6-12 MonoidalPostCoComposeMorphism[101X
  
  [33X[1;0Y[29X[2XMonoidalPostCoComposeMorphism[102X( [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,c),
            \mathrm{\underline{coHom}}(a,b)                            \otimes
            \mathrm{\underline{coHom}}(b,c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are three objects [23Xa,b,c[123X. The output is the postcocomposition
  morphism     [23X\mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}:
  \mathrm{\underline{coHom}}(a,c)  \rightarrow \mathrm{\underline{coHom}}(a,b)
  \otimes \mathrm{\underline{coHom}}(b,c)[123X.[133X
  
  [1X1.6-13 MonoidalPostCoComposeMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XMonoidalPostCoComposeMorphismWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xc[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,c),
            \mathrm{\underline{coHom}}(a,b)                            \otimes
            \mathrm{\underline{coHom}}(b,c) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs  = \mathrm{\underline{coHom}}(a,c)[123X, three
  objects  [23Xa,b,c[123X,  and  an  object [23Xr = \mathrm{\underline{coHom}}(b,c) \otimes
  \mathrm{\underline{coHom}}(a,b)[123X.   The   output   is  the  postcocomposition
  morphism     [23X\mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}:
  \mathrm{\underline{coHom}}(a,c)  \rightarrow \mathrm{\underline{coHom}}(a,b)
  \otimes \mathrm{\underline{coHom}}(b,c)[123X.[133X
  
  [1X1.6-14 CoDualOnObjects[101X
  
  [33X[1;0Y[29X[2XCoDualOnObjects[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan object[133X
  
  [33X[0;0YThe argument is an object [23Xa[123X. The output is its codual object [23Xa_{\vee}[123X.[133X
  
  [1X1.6-15 CoDualOnMorphisms[101X
  
  [33X[1;0Y[29X[2XCoDualOnMorphisms[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( b_{\vee}, a_{\vee} )[123X.[133X
  
  [33X[0;0YThe argument is a morphism [23X\alpha: a \rightarrow b[123X. The output is its codual
  morphism [23X\alpha_{\vee}: b_{\vee} \rightarrow a_{\vee}[123X.[133X
  
  [1X1.6-16 CoDualOnMorphismsWithGivenCoDuals[101X
  
  [33X[1;0Y[29X[2XCoDualOnMorphismsWithGivenCoDuals[102X( [3Xs[103X, [3Xalpha[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( b_{\vee}, a_{\vee} )[123X.[133X
  
  [33X[0;0YThe  argument is an object [23Xs = b_{\vee}[123X, a morphism [23X\alpha: a \rightarrow b[123X,
  and  an  object [23Xr = a_{\vee}[123X. The output is the dual morphism [23X\alpha_{\vee}:
  b^{\vee} \rightarrow a^{\vee}[123X.[133X
  
  [1X1.6-17 CoclosedEvaluationForCoDual[101X
  
  [33X[1;0Y[29X[2XCoclosedEvaluationForCoDual[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( 1, a_{\vee} \otimes a )[123X.[133X
  
  [33X[0;0YThe  argument is an object [23Xa[123X. The output is the coclosed evaluation morphism
  [23X\mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a[123X.[133X
  
  [1X1.6-18 CoclosedEvaluationForCoDualWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XCoclosedEvaluationForCoDualWithGivenTensorProduct[102X( [3Xs[103X, [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( 1, a_{\vee} \otimes a )[123X.[133X
  
  [33X[0;0YThe  arguments  are an object [23Xs = 1[123X, an object [23Xa[123X, and an object [23Xr = a_{\vee}
  \otimes    a[123X.    The    output   is   the   coclosed   evaluation   morphism
  [23X\mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a[123X.[133X
  
  [1X1.6-19 MorphismFromCoBidual[101X
  
  [33X[1;0Y[29X[2XMorphismFromCoBidual[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}((a_{\vee})_{\vee}, a)[123X.[133X
  
  [33X[0;0YThe  argument  is  an object [23Xa[123X. The output is the morphism from the cobidual
  [23X(a_{\vee})_{\vee} \rightarrow a[123X.[133X
  
  [1X1.6-20 MorphismFromCoBidualWithGivenCoBidual[101X
  
  [33X[1;0Y[29X[2XMorphismFromCoBidualWithGivenCoBidual[102X( [3Xa[103X, [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}((a_{\vee})_{\vee}, a)[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xa[123X, and an object [23Xs = (a_{\vee})_{\vee}[123X. The
  output is the morphism from the cobidual [23X(a_{\vee})_{\vee} \rightarrow a[123X.[133X
  
  [1X1.6-21 InternalCoHomTensorProductCompatibilityMorphism[101X
  
  [33X[1;0Y[29X[2XInternalCoHomTensorProductCompatibilityMorphism[102X( [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in [23X\mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes
            a',   b   \otimes   b'),  \mathrm{\underline{coHom}}(a,b)  \otimes
            \mathrm{\underline{coHom}}(a',b'))[123X.[133X
  
  [33X[0;0YThe  argument  is a list of four objects [23X[ a, a', b, b' ][123X. The output is the
  natural                                                             morphism
  [23X\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')[123X.[133X
  
  [1X1.6-22 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects[102X( [3Xs[103X, [3Xlist[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in [23X\mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes
            a',   b   \otimes   b'),  \mathrm{\underline{coHom}}(a,b)  \otimes
            \mathrm{\underline{coHom}}(a',b') )[123X.[133X
  
  [33X[0;0YThe arguments are a list of four objects [23X[ a, a', b, b' ][123X, and two objects [23Xs
  =   \mathrm{\underline{coHom}}(a   \otimes   a',  b  \otimes  b')[123X  and  [23Xr  =
  \mathrm{\underline{coHom}}(a,b)  \otimes  \mathrm{\underline{coHom}}(a',b')[123X.
  The          output          is         the         natural         morphism
  [23X\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')[123X.[133X
  
  [1X1.6-23 CoDualityTensorProductCompatibilityMorphism[101X
  
  [33X[1;0Y[29X[2XCoDualityTensorProductCompatibilityMorphism[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes
            b_{\vee} )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,b[123X.  The  output is the natural morphism
  [23X\mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}:     (a
  \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}[123X.[133X
  
  [1X1.6-24 CoDualityTensorProductCompatibilityMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XCoDualityTensorProductCompatibilityMorphismWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes
            b_{\vee} )[123X.[133X
  
  [33X[0;0YThe  arguments  are an object [23Xs = (a \otimes b)_{\vee}[123X, two objects [23Xa,b[123X, and
  an  object [23Xr = a_{\vee} \otimes b_{\vee}[123X. The output is the natural morphism
  [23X\mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}_{a,b}:
  (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}[123X.[133X
  
  [1X1.6-25 MorphismFromInternalCoHomToTensorProduct[101X
  
  [33X[1;0Y[29X[2XMorphismFromInternalCoHomToTensorProduct[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,b),
            b_{\vee} \otimes a )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,b[123X.  The  output is the natural morphism
  [23X\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}:
  \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a[123X.[133X
  
  [1X1.6-26 MorphismFromInternalCoHomToTensorProductWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XMorphismFromInternalCoHomToTensorProductWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,b), a
            \otimes b_{\vee} )[123X.[133X
  
  [33X[0;0YThe arguments are an object [23Xs = \mathrm{\underline{coHom}}(a,b)[123X, two objects
  [23Xa,b[123X,  and  an  object  [23Xr  =  b_{\vee}  \otimes  a[123X. The output is the natural
  morphism
  [23X\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}:
  \mathrm{\underline{coHom}}(a,b) \rightarrow a \otimes b_{\vee}[123X.[133X
  
  [1X1.6-27 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya           morphism           in           [23X\mathrm{Hom}(a_{\vee},
            \mathrm{\underline{coHom}}(1,a))[123X.[133X
  
  [33X[0;0YThe   argument   is   an   object   [23Xa[123X.   The   output   is  the  isomorphism
  [23X\mathrm{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}_{a}:
  a_{\vee} \rightarrow \mathrm{\underline{coHom}}(1,a)[123X.[133X
  
  [1X1.6-28 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya    morphism   in   [23X\mathrm{Hom}(\mathrm{\underline{coHom}}(1,a),
            a_{\vee})[123X.[133X
  
  [33X[0;0YThe   argument   is   an   object   [23Xa[123X.   The   output   is  the  isomorphism
  [23X\mathrm{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}_{a}:
  \mathrm{\underline{coHom}}(1,a) \rightarrow a_{\vee}[123X.[133X
  
  [1X1.6-29 UniversalPropertyOfCoDual[101X
  
  [33X[1;0Y[29X[2XUniversalPropertyOfCoDual[102X( [3Xt[103X, [3Xa[103X, [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a_{\vee}, t)[123X.[133X
  
  [33X[0;0YThe  arguments  are  two objects [23Xt,a[123X, and a morphism [23X\alpha: 1 \rightarrow t
  \otimes  a[123X.  The  output is the morphism [23Xa_{\vee} \rightarrow t[123X given by the
  universal property of [23Xa_{\vee}[123X.[133X
  
  [1X1.6-30 CoLambdaIntroduction[101X
  
  [33X[1;0Y[29X[2XCoLambdaIntroduction[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), 1 )[123X.[133X
  
  [33X[0;0YThe  argument  is  a  morphism  [23X\alpha:  a  \rightarrow b[123X. The output is the
  corresponding  morphism [23X \mathrm{\underline{coHom}}(a,b) \rightarrow 1[123X under
  the cohom tensor adjunction.[133X
  
  [1X1.6-31 CoLambdaElimination[101X
  
  [33X[1;0Y[29X[2XCoLambdaElimination[102X( [3Xa[103X, [3Xb[103X, [3Xalpha[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,b)[123X.[133X
  
  [33X[0;0YThe    arguments   are   two   objects   [23Xa,b[123X,   and   a   morphism   [23X\alpha:
  \mathrm{\underline{coHom}}(a,b)  \rightarrow  1[123X.  The output is a morphism [23Xa
  \rightarrow b[123X corresponding to [23X\alpha[123X under the cohom tensor adjunction.[133X
  
  [1X1.6-32 IsomorphismFromObjectToInternalCoHom[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromObjectToInternalCoHom[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1))[123X.[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output  is the natural isomorphism [23Xa
  \rightarrow \mathrm{\underline{coHom}}(a,1)[123X.[133X
  
  [1X1.6-33 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom[102X( [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1))[123X.[133X
  
  [33X[0;0YThe    argument    is    an    object    [23Xa[123X,    and    an    object    [23Xr    =
  \mathrm{\underline{coHom}}(a,1)[123X.  The  output  is  the natural isomorphism [23Xa
  \rightarrow \mathrm{\underline{coHom}}(a,1)[123X.[133X
  
  [1X1.6-34 IsomorphismFromInternalCoHomToObject[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromInternalCoHomToObject[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a)[123X.[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output  is  the  natural isomorphism
  [23X\mathrm{\underline{coHom}}(a,1) \rightarrow a[123X.[133X
  
  [1X1.6-35 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom[102X( [3Xa[103X, [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a)[123X.[133X
  
  [33X[0;0YThe    argument    is    an    object    [23Xa[123X,    and    an    object    [23Xs    =
  \mathrm{\underline{coHom}}(a,1)[123X.  The  output  is  the  natural  isomorphism
  [23X\mathrm{\underline{coHom}}(a,1) \rightarrow a[123X.[133X
  
  
  [1X1.7 [33X[0;0YSymmetric Closed Monoidal Categories[133X[101X
  
  [33X[0;0YA  monoidal  category  [23X\mathbf{C}[123X  which is symmetric and closed is called a
  [13Xsymmetric closed monoidal category[113X.[133X
  
  [33X[0;0YThe       corresponding       GAP       property       is      given      by
  [10XIsSymmetricClosedMonoidalCategory[110X.[133X
  
  
  [1X1.8 [33X[0;0YSymmetric Coclosed Monoidal Categories[133X[101X
  
  [33X[0;0YA  monoidal  category [23X\mathbf{C}[123X which is symmetric and coclosed is called a
  [13Xsymmetric coclosed monoidal category[113X.[133X
  
  [33X[0;0YThe       corresponding       GAP       property       is      given      by
  [10XIsSymmetricCoclosedMonoidalCategory[110X.[133X
  
  
  [1X1.9 [33X[0;0YRigid Symmetric Closed Monoidal Categories[133X[101X
  
  [33X[0;0YA symmetric closed monoidal category [23X\mathbf{C}[123X satisfying[133X
  
  [30X    [33X[0;6Ythe natural morphism[133X
  
  [33X[0;0Y[23X\mathrm{\underline{Hom}}(a,   a')  \otimes  \mathrm{\underline{Hom}}(b,  b')
  \rightarrow  \mathrm{\underline{Hom}}(a  \otimes  b,  a'  \otimes  b')[123X is an
  isomorphism,[133X
  
  [30X    [33X[0;6Ythe natural morphism[133X
  
  [33X[0;0Y[23Xa \rightarrow \mathrm{\underline{Hom}}(\mathrm{\underline{Hom}}(a, 1), 1)[123X is
  an isomorphism is called a [13Xrigid symmetric closed monoidal category[113X.[133X
  
  [33X[0;0YIf  no operations involving the closed structure are installed manually, the
  internal  hom  objects  will  be  derived  as  [23X\mathrm{\underline{Hom}}(a,b)
  \coloneqq a^\vee \otimes b[123X and, in particular, [23X\mathrm{\underline{Hom}}(a,1)
  \coloneqq a^\vee \otimes 1[123X.[133X
  
  [33X[0;0YThe       corresponding       GAP       property       is      given      by
  [10XIsRigidSymmetricClosedMonoidalCategory[110X.[133X
  
  [1X1.9-1 IsomorphismFromTensorProductWithDualObjectToInternalHom[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromTensorProductWithDualObjectToInternalHom[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya     morphism    in    [23X\mathrm{Hom}(    a^{\vee}    \otimes    b,
            \mathrm{\underline{Hom}}(a,b) )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,b[123X.  The  output is the natural morphism
  [23X\mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}_{a,b}:
  a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)[123X.[133X
  
  [1X1.9-2 IsomorphismFromInternalHomToTensorProductWithDualObject[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromInternalHomToTensorProductWithDualObject[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in   [23X\mathrm{Hom}(   \mathrm{\underline{Hom}}(a,b),
            a^{\vee} \otimes b )[123X.[133X
  
  [33X[0;0YThe   arguments   are  two  objects  [23Xa,b[123X.  The  output  is  the  inverse  of
  [23X\mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}[123X,     namely
  [23X\mathrm{IsomorphismFromInternalHomToTensorProductWithDualObject}_{a,b}:
  \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b[123X.[133X
  
  [1X1.9-3 MorphismFromInternalHomToTensorProduct[101X
  
  [33X[1;0Y[29X[2XMorphismFromInternalHomToTensorProduct[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in   [23X\mathrm{Hom}(   \mathrm{\underline{Hom}}(a,b),
            a^{\vee} \otimes b )[123X.[133X
  
  [33X[0;0YThe   arguments   are  two  objects  [23Xa,b[123X.  The  output  is  the  inverse  of
  [23X\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}[123X,      namely
  [23X\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}:
  \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b[123X.[133X
  
  [1X1.9-4 MorphismFromInternalHomToTensorProductWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XMorphismFromInternalHomToTensorProductWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in   [23X\mathrm{Hom}(   \mathrm{\underline{Hom}}(a,b),
            a^{\vee} \otimes b )[123X.[133X
  
  [33X[0;0YThe  arguments  are an object [23Xs = \mathrm{\underline{Hom}}(a,b)[123X, two objects
  [23Xa,b[123X,  and  an  object  [23Xr  = a^{\vee} \otimes b[123X. The output is the inverse of
  [23X\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}[123X,      namely
  [23X\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}:
  \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b[123X.[133X
  
  [1X1.9-5 TensorProductInternalHomCompatibilityMorphismInverse[101X
  
  [33X[1;0Y[29X[2XTensorProductInternalHomCompatibilityMorphismInverse[102X( [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}(  \mathrm{\underline{Hom}}(a \otimes
            b,a'    \otimes    b'),   \mathrm{\underline{Hom}}(a,a')   \otimes
            \mathrm{\underline{Hom}}(b,b') )[123X.[133X
  
  [33X[0;0YThe  argument  is a list of four objects [23X[ a, a', b, b' ][123X. The output is the
  natural                                                             morphism
  [23X\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')[123X.[133X
  
  [1X1.9-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects[102X( [3Xs[103X, [3Xlist[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya  morphism  in  [23X\mathrm{Hom}(  \mathrm{\underline{Hom}}(a \otimes
            b,a'    \otimes    b'),   \mathrm{\underline{Hom}}(a,a')   \otimes
            \mathrm{\underline{Hom}}(b,b') )[123X.[133X
  
  [33X[0;0YThe arguments are a list of four objects [23X[ a, a', b, b' ][123X, and two objects [23Xs
  =   \mathrm{\underline{Hom}}(a   \otimes   b,a'   \otimes   b')[123X   and   [23Xr  =
  \mathrm{\underline{Hom}}(a,a')  \otimes  \mathrm{\underline{Hom}}(b,b')[123X. The
  output             is             the            natural            morphism
  [23X\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')[123X.[133X
  
  [1X1.9-7 CoevaluationForDual[101X
  
  [33X[1;0Y[29X[2XCoevaluationForDual[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(1,a \otimes a^{\vee})[123X.[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output  is the coevaluation morphism
  [23X\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}[123X.[133X
  
  [1X1.9-8 CoevaluationForDualWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XCoevaluationForDualWithGivenTensorProduct[102X( [3Xs[103X, [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(1,a \otimes a^{\vee})[123X.[133X
  
  [33X[0;0YThe  arguments are an object [23Xs = 1[123X, an object [23Xa[123X, and an object [23Xr = a \otimes
  a^{\vee}[123X.  The  output  is  the  coevaluation  morphism  [23X\mathrm{coev}_{a}:1
  \rightarrow a \otimes a^{\vee}[123X.[133X
  
  [1X1.9-9 TraceMap[101X
  
  [33X[1;0Y[29X[2XTraceMap[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(1,1)[123X.[133X
  
  [33X[0;0YThe  argument  is an endomorphism [23X\alpha: a \rightarrow a[123X. The output is the
  trace morphism [23X\mathrm{trace}_{\alpha}: 1 \rightarrow 1[123X.[133X
  
  [1X1.9-10 RankMorphism[101X
  
  [33X[1;0Y[29X[2XRankMorphism[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(1,1)[123X.[133X
  
  [33X[0;0YThe   argument   is   an   object   [23Xa[123X.  The  output  is  the  rank  morphism
  [23X\mathrm{rank}_a: 1 \rightarrow 1[123X.[133X
  
  [1X1.9-11 MorphismFromBidual[101X
  
  [33X[1;0Y[29X[2XMorphismFromBidual[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}((a^{\vee})^{\vee},a)[123X.[133X
  
  [33X[0;0YThe  argument  is  an object [23Xa[123X. The output is the inverse of the morphism to
  the bidual [23X(a^{\vee})^{\vee} \rightarrow a[123X.[133X
  
  [1X1.9-12 MorphismFromBidualWithGivenBidual[101X
  
  [33X[1;0Y[29X[2XMorphismFromBidualWithGivenBidual[102X( [3Xa[103X, [3Xs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}((a^{\vee})^{\vee},a)[123X.[133X
  
  [33X[0;0YThe argument is an object [23Xa[123X, and an object [23Xs = (a^{\vee})^{\vee}[123X. The output
  is  the  inverse of the morphism to the bidual [23X(a^{\vee})^{\vee} \rightarrow
  a[123X.[133X
  
  
  [1X1.10 [33X[0;0YRigid Symmetric Coclosed Monoidal Categories[133X[101X
  
  [33X[0;0YA symmetric coclosed monoidal category [23X\mathbf{C}[123X satisfying[133X
  
  [30X    [33X[0;6Ythe natural morphism[133X
  
  [33X[0;0Y[23X\mathrm{\underline{coHom}}(a   \otimes   a',   b   \otimes  b')  \rightarrow
  \mathrm{\underline{coHom}}(a,  b) \otimes \mathrm{\underline{coHom}}(a', b')[123X
  is an isomorphism,[133X
  
  [30X    [33X[0;6Ythe natural morphism[133X
  
  [33X[0;0Y[23X\mathrm{\underline{coHom}}(1,  \mathrm{\underline{coHom}}(1, a)) \rightarrow
  a[123X is an isomorphism is called a [13Xrigid symmetric coclosed monoidal category[113X.[133X
  
  [33X[0;0YIf  no  operations  involving the coclosed structure are installed manually,
  the      internal      cohom      objects     will     be     derived     as
  [23X\mathrm{\underline{coHom}}(a,b)   \coloneqq   a   \otimes   b_\vee[123X  and,  in
  particular, [23X\mathrm{\underline{coHom}}(1,a) \coloneqq 1 \otimes a_\vee[123X.[133X
  
  [33X[0;0YThe       corresponding       GAP       property       is      given      by
  [10XIsRigidSymmetricCoclosedMonoidalCategory[110X.[133X
  
  [1X1.10-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromInternalCoHomToTensorProductWithCoDualObject[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in  [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,b),
            b_{\vee} \otimes a )[123X.[133X
  
  [33X[0;0YThe  arguments  are  two  objects  [23Xa,b[123X.  The  output is the natural morphism
  [23X\mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObjectWithGivenObjects}_{a,b}:
  \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a[123X.[133X
  
  [1X1.10-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom[101X
  
  [33X[1;0Y[29X[2XIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya     morphism    in    [23X\mathrm{Hom}(    a_{\vee}    \otimes    b,
            \mathrm{\underline{coHom}}(b,a)[123X.[133X
  
  [33X[0;0YThe   arguments   are  two  objects  [23Xa,b[123X.  The  output  is  the  inverse  of
  [23X\mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}[123X, namely
  [23X\mathrm{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}_{a,b}:
  a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)[123X.[133X
  
  [1X1.10-3 MorphismFromTensorProductToInternalCoHom[101X
  
  [33X[1;0Y[29X[2XMorphismFromTensorProductToInternalCoHom[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya     morphism    in    [23X\mathrm{Hom}(    a_{\vee}    \otimes    b,
            \mathrm{\underline{coHom}}(b,a) )[123X.[133X
  
  [33X[0;0YThe   arguments   are  two  objects  [23Xa,b[123X.  The  output  is  the  inverse  of
  [23X\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}[123X,    namely
  [23X\mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}:
  a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)[123X.[133X
  
  [1X1.10-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XMorphismFromTensorProductToInternalCoHomWithGivenObjects[102X( [3Xs[103X, [3Xa[103X, [3Xb[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya     morphism    in    [23X\mathrm{Hom}(    a_{\vee}    \otimes    b,
            \mathrm{\underline{coHom}}(b,a)[123X.[133X
  
  [33X[0;0YThe  arguments are an object [23Xs_{\vee} = a \otimes b[123X, two objects [23Xa,b[123X, and an
  object  [23Xr  =  \mathrm{\underline{coHom}}(b,a)[123X.  The output is the inverse of
  [23X\mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}[123X,    namely
  [23X\mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}:
  a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a)[123X.[133X
  
  [1X1.10-5 InternalCoHomTensorProductCompatibilityMorphismInverse[101X
  
  [33X[1;0Y[29X[2XInternalCoHomTensorProductCompatibilityMorphismInverse[102X( [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in   [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,b)
            \otimes                         \mathrm{\underline{coHom}}(a',b'),
            \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' )[123X.[133X
  
  [33X[0;0YThe  argument  is a list of four objects [23X[ a, a', b, b' ][123X. The output is the
  natural                                                             morphism
  [23X\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')[123X.[133X
  
  [1X1.10-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects[102X( [3Xs[103X, [3Xlist[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya   morphism   in   [23X\mathrm{Hom}(  \mathrm{\underline{coHom}}(a,b)
            \otimes                         \mathrm{\underline{coHom}}(a',b'),
            \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' )[123X.[133X
  
  [33X[0;0YThe arguments are a list of four objects [23X[ a, a', b, b' ][123X, and two objects [23Xs
  =  \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')[123X
  and  [23Xr  = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b')[123X. The output
  is                   the                   natural                  morphism
  [23X\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')[123X.[133X
  
  [1X1.10-7 CoclosedCoevaluationForCoDual[101X
  
  [33X[1;0Y[29X[2XCoclosedCoevaluationForCoDual[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a \otimes a_{\vee}, 1)[123X.[133X
  
  [33X[0;0YThe  argument  is  an  object  [23Xa[123X.  The  output  is the coclosed coevaluation
  morphism [23X\mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1[123X.[133X
  
  [1X1.10-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XCoclosedCoevaluationForCoDualWithGivenTensorProduct[102X( [3Xs[103X, [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a \otimes a_{\vee}, 1)[123X.[133X
  
  [33X[0;0YThe  arguments  are  an  object  [23Xs = a \otimes a_{\vee}[123X, an object [23Xa[123X, and an
  object   [23Xr   =   1[123X.   The  output  is  the  coclosed  coevaluation  morphism
  [23X\mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1[123X.[133X
  
  [1X1.10-9 CoTraceMap[101X
  
  [33X[1;0Y[29X[2XCoTraceMap[102X( [3Xalpha[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(1,1)[123X.[133X
  
  [33X[0;0YThe  argument  is an endomorphism [23X\alpha: a \rightarrow a[123X. The output is the
  cotrace morphism [23X\mathrm{cotrace}_{\alpha}: 1 \rightarrow 1[123X.[133X
  
  [1X1.10-10 CoRankMorphism[101X
  
  [33X[1;0Y[29X[2XCoRankMorphism[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(1,1)[123X.[133X
  
  [33X[0;0YThe   argument   is   an  object  [23Xa[123X.  The  output  is  the  corank  morphism
  [23X\mathrm{corank}_a: 1 \rightarrow 1[123X.[133X
  
  [1X1.10-11 MorphismToCoBidual[101X
  
  [33X[1;0Y[29X[2XMorphismToCoBidual[102X( [3Xa[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a, (a_{\vee})_{\vee})[123X.[133X
  
  [33X[0;0YThe  argument is an object [23Xa[123X. The output is the inverse of the morphism from
  the cobidual [23Xa \rightarrow (a_{\vee})_{\vee}[123X.[133X
  
  [1X1.10-12 MorphismToCoBidualWithGivenCoBidual[101X
  
  [33X[1;0Y[29X[2XMorphismToCoBidualWithGivenCoBidual[102X( [3Xa[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya morphism in [23X\mathrm{Hom}(a,(a_{\vee})_{\vee})[123X.[133X
  
  [33X[0;0YThe argument is an object [23Xa[123X, and an object [23Xr = (a_{\vee})_{\vee}[123X. The output
  is   the   inverse   of   the  morphism  from  the  cobidual  [23Xa  \rightarrow
  (a_{\vee})_{\vee}[123X.[133X
  
  
  [1X1.11 [33X[0;0YConvenience Methods[133X[101X
  
  [1X1.11-1 InternalHom[101X
  
  [33X[1;0Y[29X[2XInternalHom[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya cell[133X
  
  [33X[0;0YThis is a convenience method. The arguments are two cells [23Xa,b[123X. The output is
  the internal hom cell. If [23Xa,b[123X are two CAP objects the output is the internal
  Hom  object  [23X\mathrm{\underline{Hom}}(a,b)[123X. 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.[133X
  
  [1X1.11-2 InternalCoHom[101X
  
  [33X[1;0Y[29X[2XInternalCoHom[102X( [3Xa[103X, [3Xb[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya cell[133X
  
  [33X[0;0YThis is a convenience method. The arguments are two cells [23Xa,b[123X. The output is
  the  internal  cohom  cell.  If  [23Xa,b[123X  are  two CAP objects the output is the
  internal  cohom  object  [23X\mathrm{\underline{coHom}}(a,b)[123X. 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.[133X
  
  
  [1X1.12 [33X[0;0YAdd-methods[133X[101X
  
  [1X1.12-1 AddLeftDistributivityExpanding[101X
  
  [33X[1;0Y[29X[2XAddLeftDistributivityExpanding[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XLeftDistributivityExpanding[110X.      [23XF:      (      a,      L     )     \mapsto
  \mathtt{LeftDistributivityExpanding}(a, L)[123X.[133X
  
  [1X1.12-2 AddLeftDistributivityExpandingWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddLeftDistributivityExpandingWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XLeftDistributivityExpandingWithGivenObjects[110X.  [23XF:  (  s,  a,  L,  r ) \mapsto
  \mathtt{LeftDistributivityExpandingWithGivenObjects}(s, a, L, r)[123X.[133X
  
  [1X1.12-3 AddLeftDistributivityFactoring[101X
  
  [33X[1;0Y[29X[2XAddLeftDistributivityFactoring[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XLeftDistributivityFactoring[110X.      [23XF:      (      a,      L     )     \mapsto
  \mathtt{LeftDistributivityFactoring}(a, L)[123X.[133X
  
  [1X1.12-4 AddLeftDistributivityFactoringWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddLeftDistributivityFactoringWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XLeftDistributivityFactoringWithGivenObjects[110X.  [23XF:  (  s,  a,  L,  r ) \mapsto
  \mathtt{LeftDistributivityFactoringWithGivenObjects}(s, a, L, r)[123X.[133X
  
  [1X1.12-5 AddRightDistributivityExpanding[101X
  
  [33X[1;0Y[29X[2XAddRightDistributivityExpanding[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XRightDistributivityExpanding[110X.      [23XF:      (      L,     a     )     \mapsto
  \mathtt{RightDistributivityExpanding}(L, a)[123X.[133X
  
  [1X1.12-6 AddRightDistributivityExpandingWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddRightDistributivityExpandingWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XRightDistributivityExpandingWithGivenObjects[110X.  [23XF:  (  s,  L,  a, r ) \mapsto
  \mathtt{RightDistributivityExpandingWithGivenObjects}(s, L, a, r)[123X.[133X
  
  [1X1.12-7 AddRightDistributivityFactoring[101X
  
  [33X[1;0Y[29X[2XAddRightDistributivityFactoring[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XRightDistributivityFactoring[110X.      [23XF:      (      L,     a     )     \mapsto
  \mathtt{RightDistributivityFactoring}(L, a)[123X.[133X
  
  [1X1.12-8 AddRightDistributivityFactoringWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddRightDistributivityFactoringWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XRightDistributivityFactoringWithGivenObjects[110X.  [23XF:  (  s,  L,  a, r ) \mapsto
  \mathtt{RightDistributivityFactoringWithGivenObjects}(s, L, a, r)[123X.[133X
  
  [1X1.12-9 AddBraiding[101X
  
  [33X[1;0Y[29X[2XAddBraiding[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XBraiding[110X. [23XF: ( a, b
  ) \mapsto \mathtt{Braiding}(a, b)[123X.[133X
  
  [1X1.12-10 AddBraidingInverse[101X
  
  [33X[1;0Y[29X[2XAddBraidingInverse[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XBraidingInverse[110X. [23XF:
  ( a, b ) \mapsto \mathtt{BraidingInverse}(a, b)[123X.[133X
  
  [1X1.12-11 AddBraidingInverseWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XAddBraidingInverseWithGivenTensorProducts[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XBraidingInverseWithGivenTensorProducts[110X.   [23XF:   (   s,  a,  b,  r  )  \mapsto
  \mathtt{BraidingInverseWithGivenTensorProducts}(s, a, b, r)[123X.[133X
  
  [1X1.12-12 AddBraidingWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XAddBraidingWithGivenTensorProducts[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XBraidingWithGivenTensorProducts[110X.    [23XF:    (   s,   a,   b,   r   )   \mapsto
  \mathtt{BraidingWithGivenTensorProducts}(s, a, b, r)[123X.[133X
  
  [1X1.12-13 AddCoevaluationMorphism[101X
  
  [33X[1;0Y[29X[2XAddCoevaluationMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoevaluationMorphism[110X.  [23XF:  ( a, b ) \mapsto \mathtt{CoevaluationMorphism}(a,
  b)[123X.[133X
  
  [1X1.12-14 AddCoevaluationMorphismWithGivenRange[101X
  
  [33X[1;0Y[29X[2XAddCoevaluationMorphismWithGivenRange[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoevaluationMorphismWithGivenRange[110X.    [23XF:    (    a,    b,   r   )   \mapsto
  \mathtt{CoevaluationMorphismWithGivenRange}(a, b, r)[123X.[133X
  
  [1X1.12-15 AddDualOnMorphisms[101X
  
  [33X[1;0Y[29X[2XAddDualOnMorphisms[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XDualOnMorphisms[110X. [23XF:
  ( alpha ) \mapsto \mathtt{DualOnMorphisms}(alpha)[123X.[133X
  
  [1X1.12-16 AddDualOnMorphismsWithGivenDuals[101X
  
  [33X[1;0Y[29X[2XAddDualOnMorphismsWithGivenDuals[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XDualOnMorphismsWithGivenDuals[110X.    [23XF:    (    s,    alpha,    r   )   \mapsto
  \mathtt{DualOnMorphismsWithGivenDuals}(s, alpha, r)[123X.[133X
  
  [1X1.12-17 AddDualOnObjects[101X
  
  [33X[1;0Y[29X[2XAddDualOnObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XDualOnObjects[110X. [23XF: (
  a ) \mapsto \mathtt{DualOnObjects}(a)[123X.[133X
  
  [1X1.12-18 AddEvaluationForDual[101X
  
  [33X[1;0Y[29X[2XAddEvaluationForDual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function [23XF[123X to the category for the basic operation [10XEvaluationForDual[110X.
  [23XF: ( a ) \mapsto \mathtt{EvaluationForDual}(a)[123X.[133X
  
  [1X1.12-19 AddEvaluationForDualWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddEvaluationForDualWithGivenTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XEvaluationForDualWithGivenTensorProduct[110X.   [23XF:   (   s,   a,   r   )  \mapsto
  \mathtt{EvaluationForDualWithGivenTensorProduct}(s, a, r)[123X.[133X
  
  [1X1.12-20 AddEvaluationMorphism[101X
  
  [33X[1;0Y[29X[2XAddEvaluationMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XEvaluationMorphism[110X.
  [23XF: ( a, b ) \mapsto \mathtt{EvaluationMorphism}(a, b)[123X.[133X
  
  [1X1.12-21 AddEvaluationMorphismWithGivenSource[101X
  
  [33X[1;0Y[29X[2XAddEvaluationMorphismWithGivenSource[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XEvaluationMorphismWithGivenSource[110X.    [23XF:    (    a,    b,    s   )   \mapsto
  \mathtt{EvaluationMorphismWithGivenSource}(a, b, s)[123X.[133X
  
  [1X1.12-22 AddInternalHomOnMorphisms[101X
  
  [33X[1;0Y[29X[2XAddInternalHomOnMorphisms[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalHomOnMorphisms[110X.      [23XF:      (     alpha,     beta     )     \mapsto
  \mathtt{InternalHomOnMorphisms}(alpha, beta)[123X.[133X
  
  [1X1.12-23 AddInternalHomOnMorphismsWithGivenInternalHoms[101X
  
  [33X[1;0Y[29X[2XAddInternalHomOnMorphismsWithGivenInternalHoms[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalHomOnMorphismsWithGivenInternalHoms[110X.  [23XF:  (  s,  alpha,  beta,  r  )
  \mapsto \mathtt{InternalHomOnMorphismsWithGivenInternalHoms}(s, alpha, beta,
  r)[123X.[133X
  
  [1X1.12-24 AddInternalHomOnObjects[101X
  
  [33X[1;0Y[29X[2XAddInternalHomOnObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalHomOnObjects[110X.  [23XF:  ( a, b ) \mapsto \mathtt{InternalHomOnObjects}(a,
  b)[123X.[133X
  
  [1X1.12-25 AddInternalHomToTensorProductAdjunctionMap[101X
  
  [33X[1;0Y[29X[2XAddInternalHomToTensorProductAdjunctionMap[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalHomToTensorProductAdjunctionMap[110X.   [23XF:   (   b,   c,   g   )  \mapsto
  \mathtt{InternalHomToTensorProductAdjunctionMap}(b, c, g)[123X.[133X
  
  [1X1.12-26 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromDualObjectToInternalHomIntoTensorUnit[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromDualObjectToInternalHomIntoTensorUnit[110X.   [23XF:  (  a  )  \mapsto
  \mathtt{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}(a)[123X.[133X
  
  [1X1.12-27 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromInternalHomIntoTensorUnitToDualObject[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromInternalHomIntoTensorUnitToDualObject[110X.   [23XF:  (  a  )  \mapsto
  \mathtt{IsomorphismFromInternalHomIntoTensorUnitToDualObject}(a)[123X.[133X
  
  [1X1.12-28 AddIsomorphismFromInternalHomToObject[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromInternalHomToObject[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromInternalHomToObject[110X.      [23XF:      (      a      )     \mapsto
  \mathtt{IsomorphismFromInternalHomToObject}(a)[123X.[133X
  
  [1X1.12-29 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromInternalHomToObjectWithGivenInternalHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromInternalHomToObjectWithGivenInternalHom[110X.  [23XF: ( a, s ) \mapsto
  \mathtt{IsomorphismFromInternalHomToObjectWithGivenInternalHom}(a, s)[123X.[133X
  
  [1X1.12-30 AddIsomorphismFromObjectToInternalHom[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromObjectToInternalHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromObjectToInternalHom[110X.      [23XF:      (      a      )     \mapsto
  \mathtt{IsomorphismFromObjectToInternalHom}(a)[123X.[133X
  
  [1X1.12-31 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromObjectToInternalHomWithGivenInternalHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromObjectToInternalHomWithGivenInternalHom[110X.  [23XF: ( a, r ) \mapsto
  \mathtt{IsomorphismFromObjectToInternalHomWithGivenInternalHom}(a, r)[123X.[133X
  
  [1X1.12-32 AddLambdaElimination[101X
  
  [33X[1;0Y[29X[2XAddLambdaElimination[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function [23XF[123X to the category for the basic operation [10XLambdaElimination[110X.
  [23XF: ( a, b, alpha ) \mapsto \mathtt{LambdaElimination}(a, b, alpha)[123X.[133X
  
  [1X1.12-33 AddLambdaIntroduction[101X
  
  [33X[1;0Y[29X[2XAddLambdaIntroduction[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XLambdaIntroduction[110X.
  [23XF: ( alpha ) \mapsto \mathtt{LambdaIntroduction}(alpha)[123X.[133X
  
  [1X1.12-34 AddMonoidalPostComposeMorphism[101X
  
  [33X[1;0Y[29X[2XAddMonoidalPostComposeMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonoidalPostComposeMorphism[110X.     [23XF:     (     a,     b,    c    )    \mapsto
  \mathtt{MonoidalPostComposeMorphism}(a, b, c)[123X.[133X
  
  [1X1.12-35 AddMonoidalPostComposeMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddMonoidalPostComposeMorphismWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonoidalPostComposeMorphismWithGivenObjects[110X.  [23XF:  (  s, a, b, c, r ) \mapsto
  \mathtt{MonoidalPostComposeMorphismWithGivenObjects}(s, a, b, c, r)[123X.[133X
  
  [1X1.12-36 AddMonoidalPreComposeMorphism[101X
  
  [33X[1;0Y[29X[2XAddMonoidalPreComposeMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonoidalPreComposeMorphism[110X.     [23XF:     (     a,     b,     c    )    \mapsto
  \mathtt{MonoidalPreComposeMorphism}(a, b, c)[123X.[133X
  
  [1X1.12-37 AddMonoidalPreComposeMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddMonoidalPreComposeMorphismWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonoidalPreComposeMorphismWithGivenObjects[110X.  [23XF:  (  s,  a, b, c, r ) \mapsto
  \mathtt{MonoidalPreComposeMorphismWithGivenObjects}(s, a, b, c, r)[123X.[133X
  
  [1X1.12-38 AddMorphismFromTensorProductToInternalHom[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromTensorProductToInternalHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromTensorProductToInternalHom[110X.    [23XF:    (    a,    b    )   \mapsto
  \mathtt{MorphismFromTensorProductToInternalHom}(a, b)[123X.[133X
  
  [1X1.12-39 AddMorphismFromTensorProductToInternalHomWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromTensorProductToInternalHomWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromTensorProductToInternalHomWithGivenObjects[110X.  [23XF:  (  s, a, b, r )
  \mapsto   \mathtt{MorphismFromTensorProductToInternalHomWithGivenObjects}(s,
  a, b, r)[123X.[133X
  
  [1X1.12-40 AddMorphismToBidual[101X
  
  [33X[1;0Y[29X[2XAddMorphismToBidual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function  [23XF[123X to the category for the basic operation [10XMorphismToBidual[110X.
  [23XF: ( a ) \mapsto \mathtt{MorphismToBidual}(a)[123X.[133X
  
  [1X1.12-41 AddMorphismToBidualWithGivenBidual[101X
  
  [33X[1;0Y[29X[2XAddMorphismToBidualWithGivenBidual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismToBidualWithGivenBidual[110X.     [23XF:     (     a,     r     )     \mapsto
  \mathtt{MorphismToBidualWithGivenBidual}(a, r)[123X.[133X
  
  [1X1.12-42 AddTensorProductDualityCompatibilityMorphism[101X
  
  [33X[1;0Y[29X[2XAddTensorProductDualityCompatibilityMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductDualityCompatibilityMorphism[110X.    [23XF:    (   a,   b   )   \mapsto
  \mathtt{TensorProductDualityCompatibilityMorphism}(a, b)[123X.[133X
  
  [1X1.12-43 AddTensorProductDualityCompatibilityMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddTensorProductDualityCompatibilityMorphismWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductDualityCompatibilityMorphismWithGivenObjects[110X. [23XF: ( s, a, b, r )
  \mapsto
  \mathtt{TensorProductDualityCompatibilityMorphismWithGivenObjects}(s,  a, b,
  r)[123X.[133X
  
  [1X1.12-44 AddTensorProductInternalHomCompatibilityMorphism[101X
  
  [33X[1;0Y[29X[2XAddTensorProductInternalHomCompatibilityMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductInternalHomCompatibilityMorphism[110X.   [23XF:   (   list   )   \mapsto
  \mathtt{TensorProductInternalHomCompatibilityMorphism}(list)[123X.[133X
  
  [1X1.12-45 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddTensorProductInternalHomCompatibilityMorphismWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductInternalHomCompatibilityMorphismWithGivenObjects[110X.  [23XF: ( source,
  list,                    range                   )                   \mapsto
  \mathtt{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}(source,
  list, range)[123X.[133X
  
  [1X1.12-46 AddTensorProductToInternalHomAdjunctionMap[101X
  
  [33X[1;0Y[29X[2XAddTensorProductToInternalHomAdjunctionMap[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductToInternalHomAdjunctionMap[110X.   [23XF:   (   a,   b,   f   )  \mapsto
  \mathtt{TensorProductToInternalHomAdjunctionMap}(a, b, f)[123X.[133X
  
  [1X1.12-47 AddUniversalPropertyOfDual[101X
  
  [33X[1;0Y[29X[2XAddUniversalPropertyOfDual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XUniversalPropertyOfDual[110X.     [23XF:     (     t,     a,    alpha    )    \mapsto
  \mathtt{UniversalPropertyOfDual}(t, a, alpha)[123X.[133X
  
  [1X1.12-48 AddCoDualOnMorphisms[101X
  
  [33X[1;0Y[29X[2XAddCoDualOnMorphisms[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function [23XF[123X to the category for the basic operation [10XCoDualOnMorphisms[110X.
  [23XF: ( alpha ) \mapsto \mathtt{CoDualOnMorphisms}(alpha)[123X.[133X
  
  [1X1.12-49 AddCoDualOnMorphismsWithGivenCoDuals[101X
  
  [33X[1;0Y[29X[2XAddCoDualOnMorphismsWithGivenCoDuals[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoDualOnMorphismsWithGivenCoDuals[110X.    [23XF:   (   s,   alpha,   r   )   \mapsto
  \mathtt{CoDualOnMorphismsWithGivenCoDuals}(s, alpha, r)[123X.[133X
  
  [1X1.12-50 AddCoDualOnObjects[101X
  
  [33X[1;0Y[29X[2XAddCoDualOnObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XCoDualOnObjects[110X. [23XF:
  ( a ) \mapsto \mathtt{CoDualOnObjects}(a)[123X.[133X
  
  [1X1.12-51 AddCoDualityTensorProductCompatibilityMorphism[101X
  
  [33X[1;0Y[29X[2XAddCoDualityTensorProductCompatibilityMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoDualityTensorProductCompatibilityMorphism[110X.   [23XF:   (   a,   b   )   \mapsto
  \mathtt{CoDualityTensorProductCompatibilityMorphism}(a, b)[123X.[133X
  
  [1X1.12-52 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddCoDualityTensorProductCompatibilityMorphismWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoDualityTensorProductCompatibilityMorphismWithGivenObjects[110X. [23XF: ( s, a, b, r
  )                                                                    \mapsto
  \mathtt{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}(s,   a,
  b, r)[123X.[133X
  
  [1X1.12-53 AddCoLambdaElimination[101X
  
  [33X[1;0Y[29X[2XAddCoLambdaElimination[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoLambdaElimination[110X.      [23XF:     (     a,     b,     alpha     )     \mapsto
  \mathtt{CoLambdaElimination}(a, b, alpha)[123X.[133X
  
  [1X1.12-54 AddCoLambdaIntroduction[101X
  
  [33X[1;0Y[29X[2XAddCoLambdaIntroduction[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoLambdaIntroduction[110X.        [23XF:        (        alpha        )       \mapsto
  \mathtt{CoLambdaIntroduction}(alpha)[123X.[133X
  
  [1X1.12-55 AddCoclosedCoevaluationMorphism[101X
  
  [33X[1;0Y[29X[2XAddCoclosedCoevaluationMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoclosedCoevaluationMorphism[110X.      [23XF:      (      a,     b     )     \mapsto
  \mathtt{CoclosedCoevaluationMorphism}(a, b)[123X.[133X
  
  [1X1.12-56 AddCoclosedCoevaluationMorphismWithGivenSource[101X
  
  [33X[1;0Y[29X[2XAddCoclosedCoevaluationMorphismWithGivenSource[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoclosedCoevaluationMorphismWithGivenSource[110X.   [23XF:   (  a,  b,  s  )  \mapsto
  \mathtt{CoclosedCoevaluationMorphismWithGivenSource}(a, b, s)[123X.[133X
  
  [1X1.12-57 AddCoclosedEvaluationForCoDual[101X
  
  [33X[1;0Y[29X[2XAddCoclosedEvaluationForCoDual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoclosedEvaluationForCoDual[110X.        [23XF:       (       a       )       \mapsto
  \mathtt{CoclosedEvaluationForCoDual}(a)[123X.[133X
  
  [1X1.12-58 AddCoclosedEvaluationForCoDualWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddCoclosedEvaluationForCoDualWithGivenTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoclosedEvaluationForCoDualWithGivenTensorProduct[110X.  [23XF:  (  s, a, r ) \mapsto
  \mathtt{CoclosedEvaluationForCoDualWithGivenTensorProduct}(s, a, r)[123X.[133X
  
  [1X1.12-59 AddCoclosedEvaluationMorphism[101X
  
  [33X[1;0Y[29X[2XAddCoclosedEvaluationMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoclosedEvaluationMorphism[110X.      [23XF:      (      a,      b      )     \mapsto
  \mathtt{CoclosedEvaluationMorphism}(a, b)[123X.[133X
  
  [1X1.12-60 AddCoclosedEvaluationMorphismWithGivenRange[101X
  
  [33X[1;0Y[29X[2XAddCoclosedEvaluationMorphismWithGivenRange[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoclosedEvaluationMorphismWithGivenRange[110X.   [23XF:   (   a,   b,   r  )  \mapsto
  \mathtt{CoclosedEvaluationMorphismWithGivenRange}(a, b, r)[123X.[133X
  
  [1X1.12-61 AddInternalCoHomOnMorphisms[101X
  
  [33X[1;0Y[29X[2XAddInternalCoHomOnMorphisms[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalCoHomOnMorphisms[110X.     [23XF:     (     alpha,     beta     )     \mapsto
  \mathtt{InternalCoHomOnMorphisms}(alpha, beta)[123X.[133X
  
  [1X1.12-62 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms[101X
  
  [33X[1;0Y[29X[2XAddInternalCoHomOnMorphismsWithGivenInternalCoHoms[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalCoHomOnMorphismsWithGivenInternalCoHoms[110X.  [23XF:  (  s, alpha, beta, r )
  \mapsto  \mathtt{InternalCoHomOnMorphismsWithGivenInternalCoHoms}(s,  alpha,
  beta, r)[123X.[133X
  
  [1X1.12-63 AddInternalCoHomOnObjects[101X
  
  [33X[1;0Y[29X[2XAddInternalCoHomOnObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalCoHomOnObjects[110X.       [23XF:       (       a,      b      )      \mapsto
  \mathtt{InternalCoHomOnObjects}(a, b)[123X.[133X
  
  [1X1.12-64 AddInternalCoHomTensorProductCompatibilityMorphism[101X
  
  [33X[1;0Y[29X[2XAddInternalCoHomTensorProductCompatibilityMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalCoHomTensorProductCompatibilityMorphism[110X.   [23XF:   (   list  )  \mapsto
  \mathtt{InternalCoHomTensorProductCompatibilityMorphism}(list)[123X.[133X
  
  [1X1.12-65 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects[110X.     [23XF:    (
  source,             list,             range             )            \mapsto
  \mathtt{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}(source,
  list, range)[123X.[133X
  
  [1X1.12-66 AddInternalCoHomToTensorProductAdjunctionMap[101X
  
  [33X[1;0Y[29X[2XAddInternalCoHomToTensorProductAdjunctionMap[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalCoHomToTensorProductAdjunctionMap[110X.   [23XF:   (   a,   b,  f  )  \mapsto
  \mathtt{InternalCoHomToTensorProductAdjunctionMap}(a, b, f)[123X.[133X
  
  [1X1.12-67 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit[110X.  [23XF:  ( a ) \mapsto
  \mathtt{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}(a)[123X.[133X
  
  [1X1.12-68 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject[110X.  [23XF:  ( a ) \mapsto
  \mathtt{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}(a)[123X.[133X
  
  [1X1.12-69 AddIsomorphismFromInternalCoHomToObject[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromInternalCoHomToObject[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromInternalCoHomToObject[110X.      [23XF:      (     a     )     \mapsto
  \mathtt{IsomorphismFromInternalCoHomToObject}(a)[123X.[133X
  
  [1X1.12-70 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom[110X.  [23XF:  (  a,  s  )
  \mapsto
  \mathtt{IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom}(a, s)[123X.[133X
  
  [1X1.12-71 AddIsomorphismFromObjectToInternalCoHom[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromObjectToInternalCoHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromObjectToInternalCoHom[110X.      [23XF:      (     a     )     \mapsto
  \mathtt{IsomorphismFromObjectToInternalCoHom}(a)[123X.[133X
  
  [1X1.12-72 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom[110X.  [23XF:  (  a,  r  )
  \mapsto
  \mathtt{IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom}(a, r)[123X.[133X
  
  [1X1.12-73 AddMonoidalPostCoComposeMorphism[101X
  
  [33X[1;0Y[29X[2XAddMonoidalPostCoComposeMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonoidalPostCoComposeMorphism[110X.     [23XF:     (    a,    b,    c    )    \mapsto
  \mathtt{MonoidalPostCoComposeMorphism}(a, b, c)[123X.[133X
  
  [1X1.12-74 AddMonoidalPostCoComposeMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddMonoidalPostCoComposeMorphismWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonoidalPostCoComposeMorphismWithGivenObjects[110X.  [23XF: ( s, a, b, c, r ) \mapsto
  \mathtt{MonoidalPostCoComposeMorphismWithGivenObjects}(s, a, b, c, r)[123X.[133X
  
  [1X1.12-75 AddMonoidalPreCoComposeMorphism[101X
  
  [33X[1;0Y[29X[2XAddMonoidalPreCoComposeMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonoidalPreCoComposeMorphism[110X.     [23XF:     (     a,    b,    c    )    \mapsto
  \mathtt{MonoidalPreCoComposeMorphism}(a, b, c)[123X.[133X
  
  [1X1.12-76 AddMonoidalPreCoComposeMorphismWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddMonoidalPreCoComposeMorphismWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMonoidalPreCoComposeMorphismWithGivenObjects[110X.  [23XF:  ( s, a, b, c, r ) \mapsto
  \mathtt{MonoidalPreCoComposeMorphismWithGivenObjects}(s, a, b, c, r)[123X.[133X
  
  [1X1.12-77 AddMorphismFromCoBidual[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromCoBidual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromCoBidual[110X. [23XF: ( a ) \mapsto \mathtt{MorphismFromCoBidual}(a)[123X.[133X
  
  [1X1.12-78 AddMorphismFromCoBidualWithGivenCoBidual[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromCoBidualWithGivenCoBidual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromCoBidualWithGivenCoBidual[110X.    [23XF:    (    a,    s    )    \mapsto
  \mathtt{MorphismFromCoBidualWithGivenCoBidual}(a, s)[123X.[133X
  
  [1X1.12-79 AddMorphismFromInternalCoHomToTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromInternalCoHomToTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromInternalCoHomToTensorProduct[110X.    [23XF:    (    a,   b   )   \mapsto
  \mathtt{MorphismFromInternalCoHomToTensorProduct}(a, b)[123X.[133X
  
  [1X1.12-80 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromInternalCoHomToTensorProductWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromInternalCoHomToTensorProductWithGivenObjects[110X.  [23XF: ( s, a, b, r )
  \mapsto \mathtt{MorphismFromInternalCoHomToTensorProductWithGivenObjects}(s,
  a, b, r)[123X.[133X
  
  [1X1.12-81 AddTensorProductToInternalCoHomAdjunctionMap[101X
  
  [33X[1;0Y[29X[2XAddTensorProductToInternalCoHomAdjunctionMap[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductToInternalCoHomAdjunctionMap[110X.   [23XF:   (   c,   b,  g  )  \mapsto
  \mathtt{TensorProductToInternalCoHomAdjunctionMap}(c, b, g)[123X.[133X
  
  [1X1.12-82 AddUniversalPropertyOfCoDual[101X
  
  [33X[1;0Y[29X[2XAddUniversalPropertyOfCoDual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XUniversalPropertyOfCoDual[110X.     [23XF:     (    t,    a,    alpha    )    \mapsto
  \mathtt{UniversalPropertyOfCoDual}(t, a, alpha)[123X.[133X
  
  [1X1.12-83 AddAssociatorLeftToRight[101X
  
  [33X[1;0Y[29X[2XAddAssociatorLeftToRight[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XAssociatorLeftToRight[110X.      [23XF:      (      a,     b,     c     )     \mapsto
  \mathtt{AssociatorLeftToRight}(a, b, c)[123X.[133X
  
  [1X1.12-84 AddAssociatorLeftToRightWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XAddAssociatorLeftToRightWithGivenTensorProducts[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XAssociatorLeftToRightWithGivenTensorProducts[110X.  [23XF:  ( s, a, b, c, r ) \mapsto
  \mathtt{AssociatorLeftToRightWithGivenTensorProducts}(s, a, b, c, r)[123X.[133X
  
  [1X1.12-85 AddAssociatorRightToLeft[101X
  
  [33X[1;0Y[29X[2XAddAssociatorRightToLeft[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XAssociatorRightToLeft[110X.      [23XF:      (      a,     b,     c     )     \mapsto
  \mathtt{AssociatorRightToLeft}(a, b, c)[123X.[133X
  
  [1X1.12-86 AddAssociatorRightToLeftWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XAddAssociatorRightToLeftWithGivenTensorProducts[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XAssociatorRightToLeftWithGivenTensorProducts[110X.  [23XF:  ( s, a, b, c, r ) \mapsto
  \mathtt{AssociatorRightToLeftWithGivenTensorProducts}(s, a, b, c, r)[123X.[133X
  
  [1X1.12-87 AddLeftUnitor[101X
  
  [33X[1;0Y[29X[2XAddLeftUnitor[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function [23XF[123X to the category for the basic operation [10XLeftUnitor[110X. [23XF: ( a
  ) \mapsto \mathtt{LeftUnitor}(a)[123X.[133X
  
  [1X1.12-88 AddLeftUnitorInverse[101X
  
  [33X[1;0Y[29X[2XAddLeftUnitorInverse[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function [23XF[123X to the category for the basic operation [10XLeftUnitorInverse[110X.
  [23XF: ( a ) \mapsto \mathtt{LeftUnitorInverse}(a)[123X.[133X
  
  [1X1.12-89 AddLeftUnitorInverseWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddLeftUnitorInverseWithGivenTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XLeftUnitorInverseWithGivenTensorProduct[110X.    [23XF:    (    a,    r   )   \mapsto
  \mathtt{LeftUnitorInverseWithGivenTensorProduct}(a, r)[123X.[133X
  
  [1X1.12-90 AddLeftUnitorWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddLeftUnitorWithGivenTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XLeftUnitorWithGivenTensorProduct[110X.     [23XF:     (     a,     s     )    \mapsto
  \mathtt{LeftUnitorWithGivenTensorProduct}(a, s)[123X.[133X
  
  [1X1.12-91 AddRightUnitor[101X
  
  [33X[1;0Y[29X[2XAddRightUnitor[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XRightUnitor[110X. [23XF: ( a
  ) \mapsto \mathtt{RightUnitor}(a)[123X.[133X
  
  [1X1.12-92 AddRightUnitorInverse[101X
  
  [33X[1;0Y[29X[2XAddRightUnitorInverse[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XRightUnitorInverse[110X.
  [23XF: ( a ) \mapsto \mathtt{RightUnitorInverse}(a)[123X.[133X
  
  [1X1.12-93 AddRightUnitorInverseWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddRightUnitorInverseWithGivenTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XRightUnitorInverseWithGivenTensorProduct[110X.    [23XF:    (    a,   r   )   \mapsto
  \mathtt{RightUnitorInverseWithGivenTensorProduct}(a, r)[123X.[133X
  
  [1X1.12-94 AddRightUnitorWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddRightUnitorWithGivenTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XRightUnitorWithGivenTensorProduct[110X.     [23XF:     (     a,     s    )    \mapsto
  \mathtt{RightUnitorWithGivenTensorProduct}(a, s)[123X.[133X
  
  [1X1.12-95 AddTensorProductOnMorphisms[101X
  
  [33X[1;0Y[29X[2XAddTensorProductOnMorphisms[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductOnMorphisms[110X.     [23XF:     (     alpha,     beta     )     \mapsto
  \mathtt{TensorProductOnMorphisms}(alpha, beta)[123X.[133X
  
  [1X1.12-96 AddTensorProductOnMorphismsWithGivenTensorProducts[101X
  
  [33X[1;0Y[29X[2XAddTensorProductOnMorphismsWithGivenTensorProducts[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductOnMorphismsWithGivenTensorProducts[110X.  [23XF:  (  s, alpha, beta, r )
  \mapsto  \mathtt{TensorProductOnMorphismsWithGivenTensorProducts}(s,  alpha,
  beta, r)[123X.[133X
  
  [1X1.12-97 AddCoevaluationForDual[101X
  
  [33X[1;0Y[29X[2XAddCoevaluationForDual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoevaluationForDual[110X. [23XF: ( a ) \mapsto \mathtt{CoevaluationForDual}(a)[123X.[133X
  
  [1X1.12-98 AddCoevaluationForDualWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddCoevaluationForDualWithGivenTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoevaluationForDualWithGivenTensorProduct[110X.   [23XF:   (   s,   a,  r  )  \mapsto
  \mathtt{CoevaluationForDualWithGivenTensorProduct}(s, a, r)[123X.[133X
  
  [1X1.12-99 AddIsomorphismFromInternalHomToTensorProductWithDualObject[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromInternalHomToTensorProductWithDualObject[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromInternalHomToTensorProductWithDualObject[110X. [23XF: ( a, b ) \mapsto
  \mathtt{IsomorphismFromInternalHomToTensorProductWithDualObject}(a, b)[123X.[133X
  
  [1X1.12-100 AddIsomorphismFromTensorProductWithDualObjectToInternalHom[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromTensorProductWithDualObjectToInternalHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromTensorProductWithDualObjectToInternalHom[110X. [23XF: ( a, b ) \mapsto
  \mathtt{IsomorphismFromTensorProductWithDualObjectToInternalHom}(a, b)[123X.[133X
  
  [1X1.12-101 AddMorphismFromBidual[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromBidual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XMorphismFromBidual[110X.
  [23XF: ( a ) \mapsto \mathtt{MorphismFromBidual}(a)[123X.[133X
  
  [1X1.12-102 AddMorphismFromBidualWithGivenBidual[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromBidualWithGivenBidual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromBidualWithGivenBidual[110X.     [23XF:     (     a,     s    )    \mapsto
  \mathtt{MorphismFromBidualWithGivenBidual}(a, s)[123X.[133X
  
  [1X1.12-103 AddMorphismFromInternalHomToTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromInternalHomToTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromInternalHomToTensorProduct[110X.    [23XF:    (    a,    b    )   \mapsto
  \mathtt{MorphismFromInternalHomToTensorProduct}(a, b)[123X.[133X
  
  [1X1.12-104 AddMorphismFromInternalHomToTensorProductWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromInternalHomToTensorProductWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromInternalHomToTensorProductWithGivenObjects[110X.  [23XF:  (  s, a, b, r )
  \mapsto   \mathtt{MorphismFromInternalHomToTensorProductWithGivenObjects}(s,
  a, b, r)[123X.[133X
  
  [1X1.12-105 AddRankMorphism[101X
  
  [33X[1;0Y[29X[2XAddRankMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function [23XF[123X to the category for the basic operation [10XRankMorphism[110X. [23XF: (
  a ) \mapsto \mathtt{RankMorphism}(a)[123X.[133X
  
  [1X1.12-106 AddTensorProductInternalHomCompatibilityMorphismInverse[101X
  
  [33X[1;0Y[29X[2XAddTensorProductInternalHomCompatibilityMorphismInverse[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductInternalHomCompatibilityMorphismInverse[110X.  [23XF:  (  list ) \mapsto
  \mathtt{TensorProductInternalHomCompatibilityMorphismInverse}(list)[123X.[133X
  
  [1X1.12-107 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects[110X.  [23XF:  (
  source,             list,             range             )            \mapsto
  \mathtt{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}(source,
  list, range)[123X.[133X
  
  [1X1.12-108 AddTraceMap[101X
  
  [33X[1;0Y[29X[2XAddTraceMap[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function  [23XF[123X  to  the  category for the basic operation [10XTraceMap[110X. [23XF: (
  alpha ) \mapsto \mathtt{TraceMap}(alpha)[123X.[133X
  
  [1X1.12-109 AddCoRankMorphism[101X
  
  [33X[1;0Y[29X[2XAddCoRankMorphism[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function [23XF[123X to the category for the basic operation [10XCoRankMorphism[110X. [23XF:
  ( a ) \mapsto \mathtt{CoRankMorphism}(a)[123X.[133X
  
  [1X1.12-110 AddCoTraceMap[101X
  
  [33X[1;0Y[29X[2XAddCoTraceMap[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given  function  [23XF[123X  to the category for the basic operation [10XCoTraceMap[110X. [23XF: (
  alpha ) \mapsto \mathtt{CoTraceMap}(alpha)[123X.[133X
  
  [1X1.12-111 AddCoclosedCoevaluationForCoDual[101X
  
  [33X[1;0Y[29X[2XAddCoclosedCoevaluationForCoDual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoclosedCoevaluationForCoDual[110X.       [23XF:       (       a       )      \mapsto
  \mathtt{CoclosedCoevaluationForCoDual}(a)[123X.[133X
  
  [1X1.12-112 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct[101X
  
  [33X[1;0Y[29X[2XAddCoclosedCoevaluationForCoDualWithGivenTensorProduct[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XCoclosedCoevaluationForCoDualWithGivenTensorProduct[110X.  [23XF: ( s, a, r ) \mapsto
  \mathtt{CoclosedCoevaluationForCoDualWithGivenTensorProduct}(s, a, r)[123X.[133X
  
  [1X1.12-113 AddInternalCoHomTensorProductCompatibilityMorphismInverse[101X
  
  [33X[1;0Y[29X[2XAddInternalCoHomTensorProductCompatibilityMorphismInverse[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalCoHomTensorProductCompatibilityMorphismInverse[110X.  [23XF: ( list ) \mapsto
  \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverse}(list)[123X.[133X
  
  [1X1.12-114 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects[110X. [23XF: (
  source,             list,             range             )            \mapsto
  \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}(source,
  list, range)[123X.[133X
  
  [1X1.12-115 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromInternalCoHomToTensorProductWithCoDualObject[110X.  [23XF:  (  a,  b )
  \mapsto
  \mathtt{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}(a, b)[123X.[133X
  
  [1X1.12-116 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom[101X
  
  [33X[1;0Y[29X[2XAddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom[110X.  [23XF:  (  a,  b )
  \mapsto
  \mathtt{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}(a, b)[123X.[133X
  
  [1X1.12-117 AddMorphismFromTensorProductToInternalCoHom[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromTensorProductToInternalCoHom[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromTensorProductToInternalCoHom[110X.    [23XF:    (    a,   b   )   \mapsto
  \mathtt{MorphismFromTensorProductToInternalCoHom}(a, b)[123X.[133X
  
  [1X1.12-118 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects[101X
  
  [33X[1;0Y[29X[2XAddMorphismFromTensorProductToInternalCoHomWithGivenObjects[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismFromTensorProductToInternalCoHomWithGivenObjects[110X.  [23XF: ( s, a, b, r )
  \mapsto \mathtt{MorphismFromTensorProductToInternalCoHomWithGivenObjects}(s,
  a, b, r)[123X.[133X
  
  [1X1.12-119 AddMorphismToCoBidual[101X
  
  [33X[1;0Y[29X[2XAddMorphismToCoBidual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given function [23XF[123X to the category for the basic operation [10XMorphismToCoBidual[110X.
  [23XF: ( a ) \mapsto \mathtt{MorphismToCoBidual}(a)[123X.[133X
  
  [1X1.12-120 AddMorphismToCoBidualWithGivenCoBidual[101X
  
  [33X[1;0Y[29X[2XAddMorphismToCoBidualWithGivenCoBidual[102X( [3XC[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ynothing[133X
  
  [33X[0;0YThe  arguments  are  a  category [23XC[123X and a function [23XF[123X. This operation adds the
  given    function    [23XF[123X   to   the   category   for   the   basic   operation
  [10XMorphismToCoBidualWithGivenCoBidual[110X.     [23XF:     (    a,    r    )    \mapsto
  \mathtt{MorphismToCoBidualWithGivenCoBidual}(a, r)[123X.[133X
  
