Proposition psfunctor_fib_to_psfunctor_bicat_laws : psfunctor_laws psfunctor_fib_to_psfunctor_bicat_data. Proof. repeat split. - intros P₁ P₂ F. use modification_eq. intros x. use nat_trans_eq. { apply homset_property. } intro xx. cbn. apply idpath. - intros P₁ P₂ F G H τ θ. use modification_eq. intros x. use nat_trans_eq. { apply homset_property. } intro xx. cbn. apply maponpaths_2. apply homset_property. - intros P₁ P₂ F. use modification_eq. intros x. use nat_trans_eq. { apply homset_property. } intro xx. cbn. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. rewrite (disp_functor_id (pr1 F)). unfold transportb. rewrite !transport_f_f. refine (!_). apply transportf_set. apply homset_property. - intros P₁ P₂ F. use modification_eq. intros x. use nat_trans_eq. { apply homset_property. } intro xx. cbn. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. rewrite id_right_disp. unfold transportb. rewrite !transport_f_f. refine (!_). apply transportf_set. apply homset_property. - intros P₁ P₂ P₃ P₄ F G H. use modification_eq. intros x. use nat_trans_eq. { apply homset_property. } intro 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_left_disp. unfold transportb. rewrite transport_f_f. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. rewrite (disp_functor_id (pr1 H)). unfold transportb. rewrite !transport_f_f. apply maponpaths_2. apply homset_property. - intros P₁ P₂ P₃ F G₁ G₂ τ. use modification_eq. intros x. use nat_trans_eq. { apply homset_property. } intro xx. cbn. rewrite id_left_disp, id_right_disp. unfold transportb. rewrite !transport_f_f. apply maponpaths_2. apply homset_property. - intros P₁ P₂ P₃ F₁ F₂ G τ. use modification_eq. intros x. use nat_trans_eq. { apply homset_property. } intro xx. cbn. rewrite id_left_disp, 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.