proposition: Lemma nat_trafo_pointwise_z_iso_if_z_iso {C : precategory_data} {C' : precategory} (hs: has_homsets C') {F G : ob [C, C', hs]} (α : F --> G) : is_z_isomorphism α -> is_nat_z_iso (pr1 α).
proof: Proof. intro H. red. intro c. set (αcinv := pr1 (inv_from_z_iso (α,,H)) c). use make_is_z_isomorphism. - exact αcinv. - assert (HH := is_z_isomorphism_is_inverse_in_precat H). induction HH as [HH1 HH2]. apply (maponpaths pr1) in HH1. apply toforallpaths in HH1. apply (maponpaths pr1) in HH2. apply toforallpaths in HH2. split. + apply HH1. + apply HH2. Defined.
imports: UniMath.Foundations.Propositions UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Univalence UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.TransportMorphisms UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence