fact: Definition linear_category_laws (𝕃 : linear_category_data) : UU := (∏ (x y : 𝕃) (f : x --> y), #(linear_category_bang 𝕃) f · linear_category_comult 𝕃 y = linear_category_comult 𝕃 x · (#(linear_category_bang 𝕃) f #⊗ #(linear_category_bang 𝕃) f)) × (∏ (x y : 𝕃) (f : x --> y), #(linear_category_bang 𝕃) f · linear_category_counit 𝕃 y = linear_category_counit 𝕃 x) × (∏ (x : 𝕃), linear_category_comult 𝕃 x · (δ (linear_category_bang 𝕃) x #⊗ δ (linear_category_bang 𝕃) x) · mon_functor_tensor (linear_category_bang_functor 𝕃) _ _ = δ (linear_category_bang 𝕃) x · #(linear_category_bang 𝕃) (linear_category_comult 𝕃 x)) × (∏ (x : 𝕃), linear_category_counit 𝕃 x · mon_functor_unit (linear_category_bang_functor 𝕃) = δ (linear_category_bang 𝕃) x · #(linear_category_bang 𝕃) (linear_category_counit 𝕃 x)) × (∏ (x : 𝕃), δ (linear_category_bang 𝕃) x · linear_category_counit 𝕃 (linear_category_bang 𝕃 x) = linear_category_counit 𝕃 x) × (∏ (x : 𝕃), δ (linear_category_bang 𝕃) x · linear_category_comult 𝕃 (linear_category_bang 𝕃 x) = linear_category_comult 𝕃 x · (δ (linear_category_bang 𝕃) x #⊗ δ (linear_category_bang 𝕃) x)) × (∏ (x : 𝕃), linear_category_comult 𝕃 x · (identity _ #⊗ linear_category_comult 𝕃 x) = linear_category_comult 𝕃 x · (linear_category_comult 𝕃 x #⊗ identity _) · mon_lassociator _ _ _) × (∏ (x : 𝕃), linear_category_comult 𝕃 x · (linear_category_counit 𝕃 x #⊗ identity _) · mon_lunitor _ = identity _) × (∏ (x : 𝕃), linear_category_comult 𝕃 x · sym_mon_braiding 𝕃 _ _ = linear_category_comult 𝕃 x) × (∏ x y : 𝕃, mon_functor_tensor (linear_category_bang_functor 𝕃) x y · linear_category_comult 𝕃 (x ⊗ y) = (linear_category_comult 𝕃 x) #⊗ (linear_category_comult 𝕃 y) · (inner_swap 𝕃 (linear_category_bang 𝕃 x) (linear_category_bang 𝕃 x) (linear_category_bang 𝕃 y) (linear_category_bang 𝕃 y) · mon_functor_tensor (linear_category_bang_functor 𝕃) x y #⊗ mon_functor_tensor (linear_category_bang_functor 𝕃) x y)) × (mon_functor_unit (linear_category_bang_functor 𝕃) · linear_category_comult 𝕃 I_{𝕃} = mon_linvunitor I_{𝕃} · mon_functor_unit (linear_category_bang_functor 𝕃) #⊗ mon_functor_unit (linear_category_bang_functor 𝕃)) × (∏ x y : 𝕃, mon_functor_tensor (linear_category_bang_functor 𝕃) x y · linear_category_counit 𝕃 (x ⊗ y) = linear_category_counit 𝕃 x #⊗ linear_category_counit 𝕃 y · mon_lunitor (monoidal_unit 𝕃)) × (mon_functor_unit (linear_category_bang_functor 𝕃) · linear_category_counit 𝕃 I_{𝕃} = identity I_{𝕃}).
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monads.Comonads UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.FunctorCategories UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal UniMath.CategoryTheory.Monoidal.Structure.Closed UniMath.CategoryTheory.Monoidal.Comonoids.Category UniMath.CategoryTheory.Monoidal.Examples.ConstantFunctor UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids UniMath.CategoryTheory.Categories.Dialgebras UniMath.CategoryTheory.Monoidal.Examples.DiagonalFunctor