Lemma actor_nat1 {C : category} (Act : actegory C) {v v' w w' : V} {z z' : C} (f : V⟦v,v'⟧) (g : V⟦w,w'⟧) (h : C⟦z,z'⟧) : (actegory_actordata Act v w z) · ((f ⊗^{Act}_{r} (w ⊗_{Act} z)) · (v' ⊗^{Act}_{l} ((g ⊗^{Act}_{r} z) · (w' ⊗^{Act}_{l} h)))) = (((f ⊗^{Mon_V}_{r} w) · (v' ⊗^{Mon_V}_{l} g)) ⊗^{Act}_{r} z) · ((v' ⊗_{Mon_V} w') ⊗^{Act}_{l} h) · (actegory_actordata Act v' w' z'). Proof. rewrite assoc. rewrite (actegory_actornatright Act). rewrite assoc'. etrans. { apply cancel_precomposition. rewrite (bifunctor_leftcomp Act). rewrite assoc. rewrite (actegory_actornatleftright Act). apply idpath. } etrans. { apply cancel_precomposition. rewrite assoc'. apply cancel_precomposition. apply (actegory_actornatleft Act). } rewrite assoc. rewrite assoc. apply cancel_postcomposition. apply pathsinv0. rewrite (bifunctor_rightcomp Act). apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.Foundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. Import MonoidalNotations. Section EquivalenceFromTensorWithUnit. Import MonoidalNotations. End EquivalenceFromTensorWithUnit. Import MonoidalNotations. Import ActegoryNotations.