Definition is_invertible_2cell_to_is_nat_z_iso {C D : bicat_of_univ_cats} {F G : C --> D} (η : F ==> G) : is_invertible_2cell η → is_nat_z_iso (pr1 η). Proof. intros Hη X. use tpair. - apply (Hη^-1). - abstract (split ; cbn ; [ exact (nat_trans_eq_pointwise (vcomp_rinv Hη) X) | exact (nat_trans_eq_pointwise (vcomp_linv Hη) X)]). 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.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.catiso. Require Import UniMath.CategoryTheory.CategoryEquality. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Univalence.