Lemma is_functor_z_iso_pointwise_if_z_iso (C : precategory_data)(C' : precategory) (hs: has_homsets C') (F G : ob [C, C', hs]) (A : F --> G) : is_z_isomorphism A -> is_nat_z_iso (pr1 A). Proof. intros H a. set (T := inv_from_z_iso (tpair _ A H)). set (TA := z_iso_inv_after_z_iso (tpair _ A H)). set (TA' := z_iso_after_z_iso_inv (tpair _ A H)). simpl in *. exists (T a). split. - unfold T. set (H1' := nat_trans_eq_pointwise TA). apply H1'. - apply (nat_trans_eq_pointwise TA'). 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.