Proposition is_cartesian_equiv_over_id_comm : is_cartesian_equiv_over_id_fact ;; ♯ L ff = hh. Proof. unfold is_cartesian_equiv_over_id_fact ; cbn. rewrite mor_disp_transportf_postwhisker. rewrite assoc_disp_var. rewrite transport_f_f. etrans. { do 2 apply maponpaths. refine (!_). apply (disp_functor_comp_var L). } rewrite cartesian_factorisation_commutes. rewrite disp_functor_transportf. rewrite !mor_disp_transportf_prewhisker. rewrite !transport_f_f. rewrite disp_functor_comp. unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. etrans. { do 2 apply maponpaths. etrans. { apply maponpaths. apply triangle_1_over_id_alt. } apply (disp_nat_trans_ax ε). } unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite assoc_disp. unfold transportb. rewrite transport_f_f. etrans. { apply maponpaths. apply maponpaths_2. apply (z_iso_disp_after_inv_mor (is_z_iso_counit_over_id (pr2 HL) w ww)). } unfold transportb. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite id_left_disp. unfold transportb. rewrite transport_f_f. cbn. 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.