proposition: Proposition psfunctor_fib_to_psfunctor_bicat_laws : psfunctor_laws psfunctor_fib_to_psfunctor_bicat_data.
proof: 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.
imports: 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.opp_precat UniMath.CategoryTheory.OppositeCategory.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.IndexedCategories.IndexedCategory UniMath.CategoryTheory.IndexedCategories.IndexedFunctor UniMath.CategoryTheory.IndexedCategories.IndexedTransformation UniMath.CategoryTheory.IndexedCategories.FibrationToIndexedCategory UniMath.CategoryTheory.IndexedCategories.CartesianToIndexedFunctor UniMath.CategoryTheory.IndexedCategories.NatTransToIndexed UniMath.CategoryTheory.IndexedCategories.IndexedCategoryToFibration UniMath.CategoryTheory.IndexedCategories.IndexedFunctorToCartesian UniMath.CategoryTheory.IndexedCategories.IndexedTransformationToTransformation UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Discreteness UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Examples.DiscreteBicat UniMath.Bicategories.Core.Examples.FibSlice UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.PseudoFunctors.Examples.PseudoFunctorsIntoCat UniMath.Bicategories.Transformations.PseudoTransformation UniMath.Bicategories.Transformations.Examples.PseudoTransformationIntoCat UniMath.Bicategories.Modifications.Modification UniMath.Bicategories.Modifications.Examples.ModificationIntoCat UniMath.Bicategories.PseudoFunctors.Biequivalence