Lemma lunitorinv_preserves_leftwhiskering_with_unit {C : category} (M : monoidal C) : luinv^{M}_{I_{ M}} ⊗^{ M}_{r} I_{ M} · α^{ M }_{ I_{ M}, I_{ M}, I_{ M}} = I_{ M} ⊗^{ M}_{l} luinv^{M}_{I_{ M}}. Proof. set (t := monoidal_triangle_identity_inv_alt M I_{M} I_{M}). use (_ @ ! z_iso_inv_on_left _ _ _ _ (_,, α^{M}_{_,_,_} ,, _) _ (! t)). - apply maponpaths_2. apply maponpaths. apply unitorsinv_coincide_on_unit_alt. - split ; apply (monoidal_associatorisolaw 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.