Lemma actionbased_strength_pentagon_eq_variant2variant1 (ϛ : actionbased_strength_nat): actionbased_strength_pentagon_eq_variant2 ϛ -> actionbased_strength_pentagon_eq_variant1 ϛ. Proof. intros Heq a v w. red in Heq. apply pathsinv0. apply (z_iso_inv_to_right _ _ _ _ (make_z_iso (# F (χ ((a, v), w))) (is_z_isomorphism_mor (functor_on_is_z_isomorphism F (pr2 χ ((a, v), w)))) (functor_on_is_z_isomorphism F (pr2 χ ((a, v), w))))). etrans. { apply pathsinv0. apply Heq. } clear Heq. apply maponpaths. apply pathsinv0. apply (functor_on_inv_from_z_iso' _ (pr2 χ ((a, v), w))). 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.