proposition: Lemma precompose_inner_swap_with_augs_on_left (mx my : comonoid M) : inner_swap M mx mx my my · (ε_{ mx} ⊗^{ M} ε_{ my}) ⊗^{ M}_{r} (mx ⊗_{ M} my) = (ε_{ mx} ⊗^{M}_{r} _) #⊗ (ε_{ my} ⊗^{M}_{r} my) · inner_swap M _ _ _ _.
proof: Proof. refine (_ @ naturality_inner_swap (M) ε_{mx} (identity _) ε_{my} (identity _) @ _). - rewrite <- (when_bifunctor_becomes_rightwhiskering M). do 2 apply maponpaths. apply pathsinv0, tensor_id_id. - now rewrite <- ! (when_bifunctor_becomes_rightwhiskering M). 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.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Examples.Fullsub UniMath.CategoryTheory.Monoidal.Examples.Sigma UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Comonoids.Category