Proposition tensor_sym_mon_braiding {x₁ x₂ y₁ y₂ : V} (f : x₁ --> x₂) (g : y₁ --> y₂) : f #⊗ g · sym_mon_braiding x₂ y₂ = sym_mon_braiding x₁ y₁ · g #⊗ f. Proof. unfold monoidal_cat_tensor_mor. unfold functoronmorphisms1. rewrite !assoc'. etrans. { apply maponpaths. refine (!_). apply (pr1 (pr122 V) x₂ y₁ y₂ g). } rewrite !assoc. etrans. { apply maponpaths_2. refine (!_). apply (pr2 (pr122 V) x₁ x₂ y₁ f). } rewrite !assoc'. apply maponpaths. refine (!_). use (bifunctor_equalwhiskers V). 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.