Definition is_bifinal_to_is_bifinal_repr_help {X : B} (HX : is_bifinal X) (Y : B) : equivalence_of_cats (hom Y X) unit_category. Proof. simple refine ((_ ,, (_ ,, (_ ,, _))) ,, (_ ,, _)). - exact (functor_to_unit _). - exact (bifinal_inv HX Y). - exact (bifinal_inv_unit HX Y). - exact (bifinal_inv_counit HX Y). - intros f. cbn ; unfold bifinal_inv_unit_data. apply is_inv2cell_to_is_z_iso. apply is_bifinal_invertible_2cell_property. - intro g. cbn. apply path_univalent_groupoid. 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.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.Properties.ContainsAdjEquiv. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Univalence.