Definition is_nat_z_iso_to_is_invertible_2cell {C D : bicat_of_cats} {F G : C --> D} (η : F ==> G) : is_nat_z_iso (pr1 η) → is_invertible_2cell η. Proof. intros Hη. use tpair. - apply (nat_z_iso_inv (η ,, Hη)). - abstract (split ; [ apply nat_trans_eq ; [ apply homset_property | ] ; intros x ; cbn ; exact (z_iso_inv_after_z_iso (pr1 η x ,, _)) | apply nat_trans_eq ; [ apply homset_property | ] ; intros x ; cbn ; exact (z_iso_after_z_iso_inv (pr1 η x ,, _)) ]). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.