Proposition psfunctor_comp_fib_to_psfunctor_bicat_laws {C : category} {D₁ D₂ D₃ : disp_univalent_category C} (HD₁ : cleaving D₁) (HD₂ : cleaving D₂) (HD₃ : cleaving D₃) (F : cartesian_disp_functor (functor_identity C) D₁ D₂) (G : cartesian_disp_functor (functor_identity C) D₂ D₃) : is_modification (psfunctor_comp_fib_to_psfunctor_bicat_data HD₁ HD₂ HD₃ F G). Proof. intros x y f. use nat_trans_eq ; [ apply homset_property | ]. intros xx ; cbn. rewrite !mor_disp_transportf_postwhisker. rewrite !transport_f_f. rewrite id_left_disp. unfold transportb. rewrite !mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. rewrite id_right_disp. unfold transportb. rewrite !mor_disp_transportf_postwhisker. rewrite transport_f_f. use (cartesian_factorisation_unique (disp_functor_over_id_composite_is_cartesian (pr2 F) (pr2 G) _ _ _ _ _ _ (HD₁ _ _ _ _))). rewrite !mor_disp_transportf_postwhisker. cbn. rewrite assoc_disp_var. rewrite transport_f_f. etrans. { do 2 apply maponpaths. refine (!_). apply (disp_functor_comp_var G). } 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 cartesian_factorisation_commutes. rewrite transport_f_f. refine (!_). 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. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. apply maponpaths_2. apply homset_property. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedCategory. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedFunctor. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedTransformation. Require Import UniMath.CategoryTheory.IndexedCategories.FibrationToIndexedCategory. Require Import UniMath.CategoryTheory.IndexedCategories.CartesianToIndexedFunctor. Require Import UniMath.CategoryTheory.IndexedCategories.NatTransToIndexed. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedCategoryToFibration. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedFunctorToCartesian. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedTransformationToTransformation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Discreteness. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.DiscreteBicat. Require Import UniMath.Bicategories.Core.Examples.FibSlice. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.PseudoFunctorsIntoCat. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Transformations.Examples.PseudoTransformationIntoCat. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.Modifications.Examples.ModificationIntoCat. Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence.