Proposition isaprop_is_equiv_over_id_eq_1 (x : C) (xx : D₁ x) : η₁ x xx ;; isaprop_is_equiv_over_id_nat_z_iso x (F x xx) = transportf (λ z, _ -->[ z ] _) (!(id_right _)) (η₂ x xx). Proof. cbn. rewrite mor_disp_transportf_prewhisker. rewrite assoc_disp. unfold transportb. rewrite transport_f_f. etrans. { apply maponpaths. apply maponpaths_2. apply (disp_nat_trans_ax η₂). } unfold transportb. cbn. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite assoc_disp_var. rewrite transport_f_f. etrans. { do 2 apply maponpaths. refine (!_). apply (disp_functor_comp_var R₂). } rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. etrans. { do 3 apply maponpaths. exact (triangle_1_over_id HF₁ x xx). } unfold transportb. rewrite disp_functor_transportf. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite disp_functor_id. unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. apply maponpaths_2. 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.