proposition: Lemma whisker_to_total (x y : V) : δ_{cm (x ⊗ y)} · (x ⊗^{V}_{l} ε_{cm y}) #⊗ (ε_{cm x} ⊗^{V}_{r} y) · mon_runitor x ⊗^{V} mon_lunitor y = identity (x ⊗ y).
proof: Proof. use (z_iso_inv_to_right _ _ _ _ (_,,_)). { use (is_z_iso_bifunctor_z_iso V). - exact (_ ,, monoidal_rightunitorisolaw V x). - exact (_ ,, monoidal_leftunitorisolaw V y). } rewrite id_left. refine (_ @ id_right _). apply pathsinv0. use z_iso_inv_on_right. refine (! whisker_to_total' x y @ _). now rewrite ! assoc. 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