proposition: Proposition cartesian_disp_functor_to_indexed_functor_laws : indexed_functor_laws cartesian_disp_functor_to_indexed_functor_data.
proof: Proof. split. - intros x xx ; cbn. use (cartesian_factorisation_unique (cartesian_disp_functor_on_cartesian F (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. unfold transportb. rewrite transport_f_f. refine (!_). etrans. { refine (!_). apply (disp_functor_comp_var F). } rewrite cartesian_factorisation_commutes. rewrite disp_functor_transportf. rewrite transport_f_f. rewrite disp_functor_id. unfold transportb. rewrite transport_f_f. apply maponpaths_2. apply homset_property. - intros x y z f g xx ; cbn. rewrite mor_disp_transportf_postwhisker. use (cartesian_factorisation_unique (cartesian_disp_functor_on_cartesian F (HD₁ _ _ _ _))). rewrite !mor_disp_transportf_postwhisker. rewrite transport_f_f. 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. unfold transportb. rewrite transport_f_f. refine (!_). rewrite assoc_disp_var. rewrite transport_f_f. rewrite assoc_disp_var. rewrite transport_f_f. etrans. { do 3 apply maponpaths. refine (!_). apply (disp_functor_comp_var F). } 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. etrans. { do 2 apply maponpaths. rewrite assoc_disp. apply idpath. } unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite mor_disp_transportf_postwhisker. 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. rewrite assoc_disp_var. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. apply maponpaths_2. apply homset_property. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.IndexedCategories.IndexedCategory UniMath.CategoryTheory.IndexedCategories.IndexedFunctor UniMath.CategoryTheory.IndexedCategories.FibrationToIndexedCategory