Definition universal_coprod_cocone_has_ump_2 : bincoprod_ump_2 p. Proof. intros x u₁ u₂ ζ₁ ζ₂. use iscontraprop1. - exact (universal_coprod_cocone_has_ump_2_unique ζ₁ ζ₂). - simple refine (_ ,, _ ,, _). + exact (invmap (make_weq _ (fully_faithful_from_equivalence _ _ _ (Hp x) u₁ u₂)) (ζ₁ ,, ζ₂)). + abstract (exact (maponpaths pr1 (homotweqinvweq (make_weq _ (fully_faithful_from_equivalence _ _ _ (Hp x) u₁ u₂)) (ζ₁ ,, ζ₂)))). + abstract (exact (maponpaths dirprod_pr2 (homotweqinvweq (make_weq _ (fully_faithful_from_equivalence _ _ _ (Hp x) u₁ u₂)) (ζ₁ ,, ζ₂)))). Defined.
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.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Section UniversalFromUMP. End UniversalFromUMP.