Lemma pentagon_eq_from_alt : actionbased_strength_pentagon_eq actionA actionA' F strength_nat_from_alt. Proof. red. intros a v w. clear δ_triangle_eq. assert (Hyp := δ_pentagon_eq v w). red in Hyp. apply (maponpaths pr1) in Hyp. apply toforallpaths in Hyp. assert (Hypinst := Hyp a). clear Hyp. cbn in Hypinst. unfold strength_nat_from_alt, actionA, actionA'. cbn. do 5 rewrite id_left. do 5 rewrite id_right. assert (aux := functor_id FA' w). apply (maponpaths pr1) in aux. apply toforallpaths in aux. rewrite aux. rewrite id_right. exact Hypinst. Qed.
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.