Lemma θ_for_ab_strength_law2 : actionbased_strength_pentagon_eq Mon_endo' domain_action target_action sig θ_for_ab_strength. Proof. intros X Z' Z. apply nat_trans_eq_alt. intro c. simpl. assert (HypX := θ_Strength2_int_implies_θ_Strength2_int_nicer _ (Sig_strength_law2 sig) X Z Z'). assert (Heqc := nat_trans_eq_weq (homset_property D) _ _ HypX c). clear HypX. cbn in Heqc. etrans. { apply maponpaths_2. etrans. { apply maponpaths. apply id_right. } apply id_left. } etrans. { etrans. { apply maponpaths_2. apply functor_id. } apply id_left. } refine (!_). etrans. { do 2 apply maponpaths_2. etrans. { apply maponpaths_2. etrans. { apply maponpaths. apply functor_id. } apply functor_id. } apply id_left. } refine (!_). refine (Heqc @ _). clear Heqc. etrans. { do 2 apply maponpaths_2. apply id_left. } apply maponpaths. apply nat_trans_eq_pointwise. clear c. apply maponpaths. apply nat_trans_eq_alt. intro c. cbn. rewrite id_right. rewrite id_left. apply pathsinv0. apply functor_id. Time 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.