Lemma monoidal_pentagon_identity_inv {C : category} (M : monoidal C) (w x y z : C) : w ⊗^{ M}_{l} (αinv_{M} x y z) · αinv_{M} w (x ⊗_{ M} y) z · αinv_{M} w x y ⊗^{ M}_{r} z = αinv_{M} w x (y ⊗_{ M} z) · αinv_{M} (w ⊗_{ M} x) y z. Proof. apply pathsinv0. apply (z_iso_inv_on_right _ _ _ (z_iso_from_associator_iso M _ _ _)). unfold z_iso_from_associator_iso. unfold make_z_iso. unfold make_is_z_isomorphism. etrans. { apply (pathsinv0 (id_right _)). } apply (z_iso_inv_on_right _ _ _ (z_iso_from_associator_iso M _ _ _)). cbn. apply pathsinv0. etrans. { rewrite assoc. apply cancel_postcomposition. apply (pathsinv0 (monoidal_pentagonidentity M w x y z)). } etrans. { rewrite assoc. rewrite assoc. apply cancel_postcomposition. apply cancel_postcomposition. rewrite assoc'. apply cancel_precomposition. apply (pathsinv0 (bifunctor_leftcomp M _ _ _ _ _ _)). } etrans. { apply cancel_postcomposition. apply cancel_postcomposition. apply cancel_precomposition. apply maponpaths. apply (pr2 (z_iso_from_associator_iso M x y z)). } etrans. { apply cancel_postcomposition. apply cancel_postcomposition. apply cancel_precomposition. apply (bifunctor_leftid M). } etrans. { apply cancel_postcomposition. apply cancel_postcomposition. apply id_right. } etrans. { apply cancel_postcomposition. rewrite assoc'. apply cancel_precomposition. apply (pr2 (z_iso_from_associator_iso M w (x⊗_{M}y) z)). } etrans. { apply cancel_postcomposition. apply id_right. } etrans. { apply (pathsinv0 (bifunctor_rightcomp M _ _ _ _ _ _)). } etrans. { apply maponpaths. apply (pr2 (pr2 (z_iso_from_associator_iso M w x y))). } apply (bifunctor_rightid M). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.PartA. 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.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.opp_precat. Import BifunctorNotations. Import MonoidalNotations. Section EquivalenceFromTensorWithUnit. Import MonoidalNotations. End EquivalenceFromTensorWithUnit. Import MonoidalNotations. Import MonoidalNotations. Import MonoidalNotations. Import MonoidalNotations. Import MonoidalNotations.