Proposition isaprop_is_equiv_over_id_eq_2 (x : C) (xx : D₂ x) : ε₁ x xx = transportf (λ z, _ -->[ z ] _) (id_left _) (♯F (isaprop_is_equiv_over_id_nat_z_iso x xx) ;; ε₂ x xx). Proof. cbn. rewrite (disp_functor_transportf _ F). rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite disp_functor_comp. unfold transportb. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite assoc_disp_var. rewrite transport_f_f. refine (!_). etrans. { do 2 apply maponpaths. apply (disp_nat_trans_ax ε₂). } unfold transportb. cbn. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite assoc_disp. unfold transportb. rewrite transport_f_f. etrans. { apply maponpaths. apply maponpaths_2. exact (triangle_1_over_id HF₂ x (R₁ x xx)). } unfold transportb. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite id_left_disp. unfold transportb. rewrite transport_f_f. apply transportf_set. apply homset_property. Qed.
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.