Definition make_is_universal_coprod_cocone_essentially_surjective : essentially_surjective (universal_coprod_functor p x). Proof. intros f. use hinhpr. simple refine (_ ,, _). - exact (bincoprod_ump_1cell Hp (pr1 f) (pr2 f)). - use category_binproduct_z_iso_map. refine (_ ,, _). + use inv2cell_to_z_iso ; cbn. apply bincoprod_ump_1cell_inl. + use inv2cell_to_z_iso ; cbn. apply bincoprod_ump_1cell_inr. 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.