Definition left_adjequiv_to_left_adjequiv_repr_equiv : equivalence_of_cats (hom w x) (hom w y). Proof. use make_equivalence_of_cats. - use make_adjunction_data. + exact L. + exact R. + apply left_adjoint_to_repr_unit. + apply left_adjoint_to_repr_counit. - split. + intro f. apply is_inv2cell_to_is_z_iso ; cbn. is_iso. exact (pr2 (left_equivalence_unit_iso Hl)). + intro f. apply is_inv2cell_to_is_z_iso ; cbn. is_iso. exact (pr2 (left_equivalence_counit_iso Hl)). 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.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.EquivToAdjequiv. 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.