Lemma nat_trafo_z_iso_if_pointwise_z_iso {C : precategory_data} {C' : precategory} (hs: has_homsets C') {F G : ob [C, C', hs]} (α : F --> G) : is_nat_z_iso (pr1 α) -> is_z_isomorphism α . Proof. intro H. red. set (αinv := nat_z_iso_to_trans_inv (make_nat_z_iso _ _ α H)). exists αinv. split; apply (nat_trans_eq hs); intro c; cbn. - exact (pr1 (pr2 (H c))). - exact (pr2 (pr2 (H c))). Defined.
Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence.