proposition: Lemma cartesian_is_binproduct : cartesian = category_binproduct C C'.
proof: Proof. apply subtypePairEquality'; [ | apply isaprop_has_homsets]. apply subtypePairEquality'; [ | apply isaprop_is_precategory, has_homsets_precategory_binproduct; apply homset_property]. apply (maponpaths (tpair _ _)). use pathsdirprod. - apply funextsec. intro. apply (maponpaths (tpair _ _)). exact (transportf_set _ _ _ (isasetaprop (isasetunit _ _))). - do 5 (apply funextsec; intro). apply (maponpaths (tpair _ _)). exact (transportf_set _ _ _ (isasetaprop (isasetunit _ _))). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Categories.StandardCategories UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Examples UniMath.CategoryTheory.DisplayedCats.Examples.Reindexing UniMath.CategoryTheory.DisplayedCats.Examples.Sigma UniMath.CategoryTheory.DisplayedCats.Limits UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Fiber