Lemma iso_in_subcat_is_iso_in_precat (a b : ob (full_sub_category C C')) (f : z_iso a b): is_z_isomorphism (C:=C) (a:=pr1 a) (b:=pr1 b) (pr1 (pr1 f)). Proof. exists (pr1 (inv_from_z_iso f)). split; simpl. - set (T:=z_iso_inv_after_z_iso f). apply (base_paths _ _ T). - set (T:=z_iso_after_z_iso_inv f). apply (base_paths _ _ T). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Subcategory.Core.