proposition: Lemma whisker_to_total' (x y : V) : mon_runitor x #⊗ mon_lunitor y · δ_{cm (x ⊗ y)} · (x ⊗^{V}_{l} ε_{cm y}) #⊗ (ε_{cm x} ⊗^{V}_{r} y) = identity _.
proof: Proof. rewrite diagonal_of_tensor. etrans. { rewrite ! assoc'. do 2 apply maponpaths. apply inner_swap_before_aug. } etrans. { apply maponpaths. apply pathsinv0, tensor_comp_mor. } refine (! tensor_comp_mor _ _ _ _ @ _). rewrite comonoid_unit_law_left_inv. rewrite comonoid_unit_law_right_inv. apply tensor_id_id. 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