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. 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.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.StreetFibration.