Lemma pentagon_eq_nicer_implies_pentagon_eq_nice : pentagon_eq_nicer -> pentagon_eq_nice. Proof. intro Heq. intros X Z' Z. specialize (Heq X Z' Z). etrans. 2: { apply maponpaths_2. exact Heq. } clear Heq. repeat rewrite <- vassocr. do 2 apply maponpaths. etrans. { apply pathsinv0. apply id2_right. } apply maponpaths. etrans. 2: { apply (functor_comp(C:=hom c0 d0')(C':=hom c0 d0)). } apply pathsinv0. apply (functor_id_id (hom c0 d0') (hom c0 d0)). refine (vassocr _ _ _ @ _). etrans. { apply maponpaths_2. apply vassocl. } etrans. { apply maponpaths_2. etrans. { apply maponpaths. apply rassociator_lassociator. } apply id2_right. } rewrite rwhisker_vcomp. etrans. { apply maponpaths. apply (z_iso_after_z_iso_inv (nat_z_iso_pointwise_z_iso (strong_monoidal_functor_μ U) (Z',,Z))). } apply id2_rwhisker. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorsTensored. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.Bicategories.MonoidalCategories.PointedFunctorsMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.Actions. Require Import UniMath.Bicategories.MonoidalCategories.ConstructionOfActions. Require Import UniMath.Bicategories.MonoidalCategories.ActionOfEndomorphismsInBicat. Require Import UniMath.Bicategories.MonoidalCategories.ActionBasedStrength. Require Import UniMath.Bicategories.MonoidalCategories.MonoidalFromBicategory. Require Import UniMath.Bicategories.MonoidalCategories.ActionBasedStrongFunctorCategory. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SignatureCategory. Require Import UniMath.CategoryTheory.Core.Univalence. Import Bicat.Notations. Section Signature_From_ActionBased_Strength. End Signature_From_ActionBased_Strength. Section ActionBased_Strength_From_Signature. End ActionBased_Strength_From_Signature.