Lemma monoidal_triangle_identity''_inv {C : category} (M : monoidal C) (x y : C) : x ⊗^{M}_{l} (ruinv_{M} y) · αinv_{M} x y I_{M} = ruinv_{M} (x ⊗_{M} y). Proof. apply pathsinv0. apply (z_iso_inv_on_left _ _ _ _ ((z_iso_from_associator_iso M _ _ _))). cbn. set (ruiy := make_z_iso _ _ (monoidal_rightunitorisolaw M y)). set (ruiyx := functor_on_z_iso (leftwhiskering_functor M x) ruiy). set (ruipxy := make_z_iso _ _ (monoidal_rightunitorisolaw M (x ⊗_{ M} y))). apply pathsinv0. apply (z_iso_inv_on_right _ _ _ ruipxy). apply (z_iso_inv_on_left _ _ _ _ ruiyx). exact (! (left_whisker_with_runitor M) 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.