proposition: Lemma LNWFS_comon_structure_whiskerequals (L L' L'' : total_category (LNWFS C)) (α : fact_mor (pr1 L) (pr1 L')) (α' : fact_mor (pr1 L') (pr1 L'')) (f : arrow C) : arrow_mor11 (#(lnwfs_L_monad (pr2 L')) (lnwfs_mor (pr2 L) (pr2 L') α f)) · arrow_mor11 (lnwfs_mor (pr2 L') (pr2 L'') α' ((lnwfs_L_monad (pr2 L')) f)) = arrow_mor11 (lnwfs_mor (pr2 L') (pr2 L'') α' (lnwfs_L_monad (pr2 L) f)) · arrow_mor11 (#(lnwfs_L_monad (pr2 L'')) (lnwfs_mor (pr2 L) (pr2 L') α f)).
proof: Proof. set (α'nat := nat_trans_ax α' _ _ (lnwfs_mor (pr2 L) (pr2 L') α f)). set (α'nat11 := base_paths _ _ (fiber_paths α'nat)). apply pathsinv0. etrans. exact (pathsinv0 α'nat11). etrans. apply pr1_transportf_const. reflexivity. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.Graphs.Coequalizers UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.EqDiag UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Chains.Chains UniMath.CategoryTheory.slicecat UniMath.CategoryTheory.Monads.Monads UniMath.CategoryTheory.Monads.MonadAlgebras UniMath.CategoryTheory.Monads.Comonads UniMath.CategoryTheory.Monads.ComonadCoalgebras UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.DisplayedCats.Examples.Arrow UniMath.CategoryTheory.DisplayedCats.Examples.Three UniMath.ModelCategories.NWFS UniMath.ModelCategories.Generated.MonoidalHelpers UniMath.ModelCategories.Helpers UniMath.ModelCategories.Generated.MonoidalHelpers UniMath.ModelCategories.Generated.FFMonoidalStructure