proposition: Lemma cartesian_monoidal_has_unique_comonoids : ∏ x : V, iscontr (disp_cat_of_comonoids V x).
proof: Proof. intro x. exists (comonoid_to_struct V (cartesian_monoidal_has_enough_comonoids x)). intro m. use subtypePath. { intro ; apply isaprop_comonoid_laws. } apply dirprodeq. - use (BinProductArrowUnique _ _ _ (is_cartesian_BinProduct Ccart x x)). + refine (_ @ comonoid_to_law_unit_right _ (_,,m)). cbn. unfold semi_cart_tensor_pr1. rewrite ! assoc. apply maponpaths_2. apply maponpaths. unfold monoidal_cat_tensor_mor. rewrite (when_bifunctor_becomes_leftwhiskering M). apply maponpaths. apply (semi_cart_to_unit_eq Ccart). + refine (_ @ comonoid_to_law_unit_left _ (_,,m)). cbn. unfold semi_cart_tensor_pr2. rewrite ! assoc. apply maponpaths_2. apply maponpaths. unfold monoidal_cat_tensor_mor. rewrite (when_bifunctor_becomes_rightwhiskering M). apply maponpaths. apply (semi_cart_to_unit_eq Ccart). - apply (semi_cart_to_unit_eq Ccart). 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.catiso UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Projection UniMath.CategoryTheory.DisplayedCats.CatIsoDisplayed 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 UniMath.CategoryTheory.Monoidal.Comonoids.Monoidal UniMath.CategoryTheory.Monoidal.Comonoids.Symmetric