Lemma signature_from_ab_strength_law1 : θ_Strength1_int θ_for_signature. Proof. red. intro X. apply nat_trans_eq_alt. intro c. cbn. assert (HypX := triangle_eq_implies_triangle_eq_nice _ _ _ _ _ _ (ab_strength_triangle _ ab_str) X). assert (Heqc := nat_trans_eq_weq (homset_property D) _ _ HypX c). cbn in Heqc. intermediate_path (# (pr1 (H X)) (id₁ c) · id₁ (pr1(pr1 H X) c)). 2: { etrans. { apply id_right. } apply (functor_id (H X)). } etrans. 2: { exact Heqc. } clear HypX Heqc. apply maponpaths. apply nat_trans_eq_pointwise. clear c. apply maponpaths. apply nat_trans_eq_alt. intro c. cbn. apply pathsinv0. etrans. { apply id_right. } apply functor_id. 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.