Definition make_is_universal_coprod_cocone (HB_2_1 : is_univalent_2_1 B) (p : bincoprod_cocone) (Hp : has_bincoprod_ump p) : is_universal_coprod_cocone p. Proof. intro x. use rad_equivalence_of_cats. - apply is_univ_hom. exact HB_2_1. - use full_and_faithful_implies_fully_faithful. split. + apply make_is_universal_coprod_cocone_full. apply Hp. + apply make_is_universal_coprod_cocone_faithful. apply Hp. - apply make_is_universal_coprod_cocone_essentially_surjective. apply Hp. 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.