proposition: Lemma DH0_DH1_is_disp_nat_z_iso {DH0 DH1 :disp_functor H D' D''} (β: disp_nat_z_iso (DF ◦ DH0) (DF ◦ DH1) (nat_z_iso_id (F ∙ H))) : is_disp_nat_z_iso (nat_z_iso_id H) (DH0_DH1_nat_trans β).
proof: Proof. set (β_inv := disp_nat_z_iso_to_trans_inv β). intros c' d'. use tpair. - cbn. apply (DF_inv_mor DH1 DH0 c' d'). exact (β_inv ↙d' ←d'). - split. * apply DF_inv_mor_simpl. etrans. apply (pr2 (pr2 β ↙d' ←d')). apply (maponpaths (λ e, transportf _ e _)). apply uip. apply homset_property. * apply DF_inv_mor_simpl. etrans. apply (pr2 (pr2 β ↙d' ←d')). apply (maponpaths (λ e, transportf _ e _)). apply uip. apply homset_property. Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.StreetFibration