proposition: Lemma make_is_binprod_from_comonoids_uniqueness (f : V⟦z, x ⊗ y⟧) (px : f · (identity x #⊗ εy · mon_runitor x) = fx) (py : f · (εx #⊗ identity y · mon_lunitor y) = fy) : f = δz · fx #⊗ fy.
proof: Proof. rewrite <- px. rewrite <- py. clear px py. etrans. 2: { apply maponpaths. apply pathsinv0. apply tensor_comp_mor. } etrans. 2: { do 2 apply maponpaths. apply pathsinv0. apply tensor_comp_mor. } rewrite assoc. etrans. 2: { apply maponpaths_2. apply pathsinv0. apply (mf _ _ f). } refine (! id_right _ @ _). rewrite assoc'. apply maponpaths. apply p. 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