proposition: Proposition fiber_functor_natural_laws : is_nat_trans _ _ fiber_functor_natural_data.
proof: 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.
imports: UniMath.Foundations.Sets UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.MonoEpiIso UniMath.CategoryTheory.Categories.HSET.Univalence UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Presheaf UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Examples.Opposite