Proposition fiber_functor_natural_nat_z_iso_eq {C₁ C₂ : category} {F : C₁ ⟶ C₂} {D₁ : disp_cat C₁} {D₂ : disp_cat C₂} (HD₁ : cleaving D₁) (HD₂ : cleaving D₂) (FF : disp_functor F D₁ D₂) (HFF₁ HFF₂ : is_cartesian_disp_functor FF) {x y : C₁} (f : y --> x) : fiber_functor_natural_nat_z_iso HD₁ HD₂ (FF ,, HFF₁) f = fiber_functor_natural_nat_z_iso HD₁ HD₂ (FF ,, HFF₂) f. Proof. use subtypePath. { intro. apply isaprop_is_nat_z_iso. } use nat_trans_eq_alt. intro xx ; cbn. use (cartesian_factorisation_unique (HFF₁ _ _ _ _ _ _ (HD₁ x y f xx))). rewrite !cartesian_factorisation_commutes. apply idpath. 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.