Proposition fiber_functor_natural_laws : is_nat_trans _ _ fiber_functor_natural_data. Proof. intros xx yy ff. unfold fiber_functor_natural_data ; cbn. use (cartesian_factorisation_unique (cartesian_disp_functor_on_cartesian FF (HD₁ _ _ _ _))). 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. refine (!_). rewrite assoc_disp_var. rewrite !mor_disp_transportf_prewhisker. rewrite !mor_disp_transportf_postwhisker. rewrite !transport_f_f. etrans. { do 3 apply maponpaths. refine (!_). apply (disp_functor_comp_var FF). } rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. 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. rewrite assoc_disp. unfold transportb. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite mor_disp_transportf_postwhisker. rewrite !transport_f_f. 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.