Lemma sym_mon_tensor_lassociator1 (x y z : V) : mon_lassociator y x z · y ⊗^{V}_{l} sym_mon_braiding x z = sym_mon_braiding y x ⊗^{V}_{r} z · mon_lassociator x y z · sym_mon_braiding x (y ⊗ z) · mon_lassociator y z x. Proof. apply pathsinv0. etrans. { rewrite assoc'. apply maponpaths. exact (sym_mon_tensor_lassociator0 x y z). } etrans. { rewrite ! assoc. do 3 apply maponpaths_2. rewrite assoc'. apply maponpaths. apply mon_lassociator_rassociator. } rewrite id_right. rewrite <- (when_bifunctor_becomes_leftwhiskering V). rewrite <- (when_bifunctor_becomes_rightwhiskering V). etrans. { do 2 apply maponpaths_2. rewrite (when_bifunctor_becomes_rightwhiskering V). apply maponpaths. apply (when_bifunctor_becomes_rightwhiskering V). } etrans. { do 2 apply maponpaths_2. apply pathsinv0, (bifunctor_rightcomp V). } rewrite sym_mon_braiding_inv. rewrite bifunctor_rightid. now rewrite id_left. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. Import MonoidalNotations.