Definition locally_posetal_left_adj_equiv_weq_z_iso {B : bicat} (HB : is_locally_posetal_bicat B) (x y : B) : @z_iso (locally_posetal_bicat_to_category HB) x y ≃ adjoint_equivalence x y. Proof. use make_weq. - exact (λ f, _ ,, z_iso_to_locally_posetal_left_adj_equiv HB (pr2 f)). - use isweq_iso. + exact (λ f, make_z_iso' _ (locally_posetal_left_adj_equiv_to_z_iso HB f)). + abstract (intro Hf ; use subtypePath ; [ intro ; apply isaprop_is_z_isomorphism | ] ; apply idpath). + abstract (intro Hf ; use subtypePath ; [ intro ; apply isaprop_left_adjoint_equivalence ; apply HB | ] ; apply idpath). 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.