Definition z_iso_to_locally_posetal_left_adj_equiv {B : bicat} (HB : is_locally_posetal_bicat B) {x y : B} {f : x --> y} (Hf : @is_z_isomorphism (locally_posetal_bicat_to_category HB) x y f) : left_adjoint_equivalence f. Proof. simple refine ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _))). - exact (inv_from_z_iso (make_z_iso' _ Hf)). - abstract (apply idtoiso_2_1 ; refine (!_) ; exact (z_iso_inv_after_z_iso (make_z_iso' _ Hf))). - abstract (apply idtoiso_2_1 ; exact (z_iso_after_z_iso_inv (make_z_iso' _ Hf))). - apply HB. - apply HB. - use make_is_invertible_2cell. + abstract (apply idtoiso_2_1 ; exact (z_iso_inv_after_z_iso (make_z_iso' _ Hf))). + apply HB. + apply HB. - use make_is_invertible_2cell. + abstract (apply idtoiso_2_1 ; refine (!_) ; exact (z_iso_after_z_iso_inv (make_z_iso' _ Hf))). + apply HB. + apply HB. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique.