Lemma monoidal_triangle_identity'_inv (x y : C) : luinv_{M} (x ⊗_{M} y) · αinv_{M} I_{M} x y = luinv_{M} x ⊗^{M}_{r} y. Proof. apply pathsinv0. apply (z_iso_inv_on_left _ _ _ _ ((z_iso_from_associator_iso M _ _ _))). cbn. set (luix := make_z_iso _ _ (monoidal_leftunitorisolaw M x)). set (luixy := functor_on_z_iso (rightwhiskering_functor M y) luix). set (luipxy := make_z_iso _ _ (monoidal_leftunitorisolaw M (x ⊗_{ M} y))). apply pathsinv0. apply (z_iso_inv_on_right _ _ _ luixy). apply (z_iso_inv_on_left _ _ _ _ luipxy). exact (! monoidal_triangleidentity' x y). 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.