Proposition sym_mon_braiding_lunitor (x : V) : sym_mon_braiding x (I_{V}) · mon_lunitor x = mon_runitor x. Proof. refine (!(id_right _) @ _ @ id_left _). rewrite <- mon_rinvunitor_runitor. rewrite !assoc. apply maponpaths_2. rewrite tensor_rinvunitor. rewrite tensor_comp_id_r. rewrite sym_mon_tensor_lassociator'. rewrite !assoc'. rewrite <- mon_lunitor_triangle. etrans. { do 5 apply maponpaths. rewrite !assoc. rewrite mon_rassociator_lassociator. apply id_left. } rewrite <- mon_rinvunitor_triangle. rewrite !assoc'. etrans. { apply maponpaths. rewrite !assoc. rewrite mon_rassociator_lassociator. rewrite id_left. apply idpath. } rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. refine (_ @ sym_mon_braiding_inv _ _). apply maponpaths. rewrite tensor_lunitor. rewrite !assoc. refine (_ @ id_left _). apply maponpaths_2. rewrite !assoc'. rewrite mon_lunitor_triangle. rewrite <- tensor_comp_id_r. rewrite mon_lunitor_I_mon_runitor_I. rewrite mon_rinvunitor_runitor. apply tensor_id_id. 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.