Definition disp_functor_eq_step_5 : (∑ (p : ∏ (x : C₁) (xx : D₁ x), z_iso_disp (identity_z_iso _) (FF x xx) (GG x xx)), ∏ (x y : C₁) (xx : D₁ x) (yy : D₁ y) (f : x --> y) (ff : xx -->[ f ] yy), ♯FF ff ;; p y yy = transportb _ (id_right _ @ !(id_left _)) (p x xx ;; ♯GG ff)) ≃ disp_nat_z_iso FF GG (nat_z_iso_id F). Proof. use weq_iso. - exact disp_functor_eq_step_5_left. - exact disp_functor_eq_step_5_right. - intro p. use subtypePath. { intro. repeat (use impred ; intro). apply D₂. } use funextsec ; intro x. use funextsec ; intro xx. use subtypePath. { intro. apply isaprop_is_z_iso_disp. } apply idpath. - intro p. use subtypePath. { intro. repeat (use impred ; intro). apply isaprop_is_z_iso_disp. } use disp_nat_trans_eq. intros x xx. apply idpath. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. 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.