Definition ab_strength_identity_functor {A : category} (actn : action Mon_V A) : actionbased_strength actn actn (functor_identity A). Proof. use tpair. - use make_nat_trans. + intro av. apply identity. + intros av av' fg. cbn. rewrite id_left. apply id_right. - split. + intro a. cbn. apply id_left. + intros a v w. cbn. rewrite binprod_id. do 2 rewrite id_right. etrans. 2: {apply cancel_postcomposition. apply pathsinv0, functor_id. } apply pathsinv0, id_left. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.Bicategories.MonoidalCategories.EndofunctorsMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.Actions. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorsTensored. Require Import UniMath.Bicategories.MonoidalCategories.EndofunctorsWhiskeredMonoidal. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Import MonoidalNotations. Import BifunctorNotations. Import MonoidalNotations.