proposition: Lemma linear_category_comult_factors_through_comult_bang : δ (linear_category_bang 𝕃) x · (linear_category_comult 𝕃 (bang x) · ε bang (bang x) #⊗ ε bang (bang x)) = linear_category_comult 𝕃 x.
proof: Proof. rewrite assoc. etrans. { apply maponpaths_2. apply linear_category_comult_comonoid_mor_comult. } rewrite assoc'. rewrite <- tensor_comp_mor. refine (_ @ id_right _). apply maponpaths. rewrite <- tensor_id_id. use two_arg_paths ; apply Comonad_law1. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Retracts UniMath.CategoryTheory.Monads.Comonads UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Adjunctions UniMath.CategoryTheory.Monoidal.FunctorCategories UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.Closed UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.Categories.Dialgebras UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Comonoids.Category UniMath.CategoryTheory.Monoidal.Comonoids.MonoidalCartesianBuilder UniMath.CategoryTheory.Monoidal.Comonoids.TransportComonoidAlongRetraction UniMath.CategoryTheory.Categories.CoEilenbergMoore UniMath.CategoryTheory.Monoidal.Examples.MonoidalDialgebras UniMath.CategoryTheory.Monoidal.Examples.SymmetricMonoidalCoEilenbergMoore UniMath.Semantics.LinearLogic.LinearCategory UniMath.Semantics.LinearLogic.LinearCategoryEilenbergMooreAdjunction UniMath.Semantics.LinearLogic.LinearNonLinear