proposition: Lemma inner_swap_along_unit (x y : V) : inner_swap x I_{V} I_{V} y = identity _.
proof: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Structure.Symmetric