Proposition isaprop_is_equiv_over_id_is_nat_z_iso : is_disp_nat_z_iso (nat_z_iso_id (functor_identity C)) isaprop_is_equiv_over_id_nat_trans. Proof. intros x xx ; cbn. pose (η₂ x (R₁ x xx) ;; ♯ R₂ (ε₁ x xx)). cbn in m. use (@is_z_iso_disp_transportf_fun_eq C D₁ x x (z_iso_comp (identity_z_iso x) (identity_z_iso x))). use (z_iso_disp_comp (_ ,, _) (_ ,, _)). - apply HF₂. - exact (is_z_iso_disp_independent_of_is_z_iso _ _ (pr2 (disp_functor_on_z_iso_disp R₂ (make_z_iso_disp _ (is_z_iso_counit_over_id HF₁ x xx))))). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. 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.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq.