Proposition sym_mon_tensor_lassociator' (x y z : V) : sym_mon_braiding x y #⊗ identity z = mon_lassociator x y z · sym_mon_braiding x (y ⊗ z) · mon_lassociator y z x · identity y #⊗ sym_mon_braiding z x · mon_rassociator y x z. Proof. rewrite (sym_mon_hexagon_lassociator x y z). rewrite !assoc'. refine (!(id_right _) @ _). apply maponpaths. refine (!_). etrans. { apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_l. rewrite sym_mon_braiding_inv. rewrite tensor_id_id. apply id_left. } apply mon_lassociator_rassociator. 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.