proposition: Lemma pentagon_eq_from_alt : actionbased_strength_pentagon_eq actionA actionA' F strength_nat_from_alt.
proof: 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.
imports: UniMath.Foundations.PartD UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.HorizontalComposition UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.Bicategories.MonoidalCategories.EndofunctorsMonoidal UniMath.Bicategories.MonoidalCategories.Actions UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorsTensored UniMath.Bicategories.MonoidalCategories.EndofunctorsWhiskeredMonoidal UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfCats