Proposition fiber_functor_natural_inv_left (xx : D₁ x) : fiber_functor_natural xx · fiber_functor_natural_inv xx = identity _. Proof. cbn. unfold fiber_functor_natural_data, fiber_functor_natural_inv ; cbn. use (cartesian_factorisation_unique (HD₂ _ _ _ _)). rewrite id_left_disp. rewrite mor_disp_transportf_postwhisker. rewrite assoc_disp_var. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite transport_f_f. unfold transportb. apply maponpaths_2. apply homset_property. Qed.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Opposite. Section IsoCleavingFromOpcleaving. End IsoCleavingFromOpcleaving.