Definition roundtrip2_ob_nat_trans_data : nat_trans_data (ActionBasedStrongFunctorCategoryToSignatureCategory ∙ SignatureCategoryToActionBasedStrongFunctorCategory) (functor_identity (Strong_Functor_category Mon_endo' domain_action target_action)). Proof. intro FF. cbn. use tpair. - use make_nat_trans. + intro X. exact (identity (pr1 (pr1 FF) X)). + intros X1 X2 f. etrans. { apply (id_right (#(pr11 FF) f)). } etrans. 2: { apply pathsinv0. apply (id_left (#(pr11 FF) f)). } apply idpath. - intros X Y. red. apply nat_trans_eq_alt. intro c. etrans. { apply id_right. } etrans. 2: { apply cancel_postcomposition. cbn. apply pathsinv0. apply id_right. } etrans. 2: { apply cancel_postcomposition. apply pathsinv0. apply (functor_id (pr11 FF X)). } apply pathsinv0. apply id_left. Defined.
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.