Lemma lunitor_preserves_leftwhiskering_with_unit : lu^{M}_{I_{ M} ⊗_{M} I_{M}} = I_{M} ⊗^{ M}_{l} lu^{M}_{I_{ M}}. Proof. apply pathsinv0. set (lun := monoidal_leftunitornat M _ _ (lu_{M} (I_{M}))). etrans. { apply (! id_right _). } etrans. 2: { apply id_right. } etrans. { apply maponpaths. exact (! pr1 (monoidal_leftunitorisolaw M I_{ M})). } etrans. { apply assoc. } etrans. { apply maponpaths_2 ; exact lun. } etrans. { apply assoc'. } apply maponpaths. apply monoidal_leftunitorisolaw. 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.