Definition make_is_universal_equifier_cone (HB_2_1 : is_univalent_2_1 B) (cone : equifier_cone) (H : has_equifier_ump cone) : is_univeral_equifier_cone cone. 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_equifier_cone_full. exact H. + apply make_is_universal_equifier_cone_faithful. exact H. - apply make_is_universal_equifier_cone_essentially_surjective. exact H. 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.Subcategory.Core. Require Import UniMath.CategoryTheory.Subcategory.Full. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. 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. Require Import UniMath.Bicategories.Morphisms.FullyFaithful.