Definition Products_category_of_posets (J : UU) : Products J category_of_posets. Proof. intro D. use make_Product. - exact (_ ,, depfunction_poset _ (λ j, pr2 (D j))). - exact (λ j, _ ,, is_monotone_depfunction_poset_pr _ _ j). - intros R f. use iscontraprop1. + abstract (use invproofirrelevance ; intros φ₁ φ₂ ; use subtypePath ; [ intro ; use impred ; intro ; apply homset_property | ] ; use eq_monotone_function ; intro x ; use funextsec ; intro j ; exact (eqtohomot (maponpaths pr1 (pr2 φ₁ j @ !(pr2 φ₂ j))) x)). + simple refine (_ ,, _). * exact (_ ,, is_monotone_depfunction_poset_pair (λ j, pr1 (f j)) (λ j, pr2 (f j))). * abstract (intro j ; use eq_monotone_function ; cbn ; intro x ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.SIP. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.DisplayedCats.Binproducts. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Categories.HSET.All.