Definition forgetful_functor_from_ptd_as_strong_monoidal_functor_alt : strong_monoidal_functor (monoidal_cat_of_pointedfunctors C) (monoidal_cat_from_bicat_and_ob (C:=bicat_of_cats) C). Proof. use tpair. - apply (make_lax_monoidal_functor (monoidal_cat_of_pointedfunctors C) (monoidal_cat_from_bicat_and_ob (C:=bicat_of_cats) C) functor_ptd_forget_alt (nat_trans_id _) aux). + abstract (intros PF1 PF2 PF3 ; apply nat_trans_eq_alt ; intro c ; cbn ; do 2 rewrite functor_id ; repeat rewrite id_right ; apply functor_id). + abstract (intro PF ; split; apply nat_trans_eq_alt; intro c; cbn ; [ do 3 rewrite id_right ; apply pathsinv0 ; apply functor_id | do 3 rewrite id_right ; apply idpath]). - split ; [ apply (nat_trafo_z_iso_if_pointwise_z_iso (homset_property _)); apply is_nat_z_iso_nat_trans_id | red ; intro c ; exists (nat_trans_id _) ; split; cbn ; [ apply nat_trans_eq; [apply homset_property |] ; intro c'; cbn ; apply id_left | apply nat_trans_eq; [apply homset_property |]; intro c'; cbn ; 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.