proposition: Lemma inner_swap_before_aug (x y : V) : inner_swap V x x y y · (x ⊗^{V}_{l} ε_{cm y}) #⊗ (ε_{cm _} ⊗^{V}_{r} y) = (_ ⊗^{V}_{l} ε_{cm _}) #⊗ (ε_{cm _} ⊗^{V}_{r} _).
proof: Proof. refine (_ @ naturality_inner_swap V (identity x) ε_{cm x} ε_{cm y} (identity y) @ _). { now rewrite <- (when_bifunctor_becomes_leftwhiskering V), <- (when_bifunctor_becomes_rightwhiskering V). } rewrite <- (when_bifunctor_becomes_leftwhiskering V), <- (when_bifunctor_becomes_rightwhiskering V). rewrite inner_swap_along_unit. apply id_right. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal UniMath.CategoryTheory.Monoidal.Comonoids.Category UniMath.CategoryTheory.Monoidal.Comonoids.Tensor