Lemma inner_swap_along_unit (x y : V) : inner_swap x I_{V} I_{V} y = identity _. Proof. unfold inner_swap. rewrite sym_mon_braiding_id. rewrite (bifunctor_rightid V). rewrite id_left. etrans. { apply maponpaths. apply maponpaths_2. apply maponpaths. apply monoidal_associatorisolaw. } rewrite (bifunctor_leftid V). rewrite id_left. apply monoidal_associatorisolaw. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import MonoidalNotations.