Lemma cartesian_is_binproduct : cartesian = category_binproduct C C'. 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.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Limits.Graphs.Limits. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Examples. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Reindexing. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Sigma. Require Import UniMath.CategoryTheory.DisplayedCats.Limits. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber.