Proposition is_pstrans_psfunctor_fib_to_psfunctor_unit : is_pstrans psfunctor_fib_to_psfunctor_unit_data. Proof. refine (_ ,, _ ,, _). - intros P₁ P₂ F G τ. use disp_nat_trans_eq. intros x xx. cbn ; unfold psfunctor_fib_to_psfunctor_unit_natural_data. rewrite mor_disp_transportf_prewhisker. rewrite mor_disp_transportf_postwhisker. rewrite mor_disp_transportf_prewhisker. rewrite !transport_f_f. rewrite id_right_disp. rewrite id_left_disp. unfold transportb. 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 id_right_disp. unfold transportb. rewrite transport_f_f. apply maponpaths_2. apply homset_property. - intros P. use disp_nat_trans_eq. intros x xx. cbn ; unfold psfunctor_fib_to_psfunctor_unit_natural_data. rewrite transportf_object_cartesian_lift. unfold transportb. rewrite !mor_disp_transportf_prewhisker. rewrite !mor_disp_transportf_postwhisker. rewrite !transport_f_f. rewrite !id_left_disp. unfold transportb. rewrite !mor_disp_transportf_postwhisker. rewrite !mor_disp_transportf_prewhisker. rewrite !transport_f_f. rewrite !id_left_disp. unfold transportb. rewrite !transport_f_f. refine (!_). rewrite assoc_disp_var. rewrite !transport_f_f. rewrite assoc_disp_var. rewrite !transport_f_f. rewrite cartesian_factorisation_commutes. rewrite !transport_f_f. rewrite cartesian_factorisation_commutes. rewrite !mor_disp_transportf_prewhisker. rewrite !transport_f_f. etrans. { do 2 apply maponpaths. rewrite assoc_disp. unfold transportb. rewrite cartesian_factorisation_commutes. rewrite !mor_disp_transportf_postwhisker. rewrite transport_f_f. rewrite assoc_disp_var. rewrite cartesian_factorisation_commutes. rewrite transport_f_f. apply idpath. } rewrite !mor_disp_transportf_prewhisker. rewrite !transport_f_f. rewrite id_right_disp. unfold transportb. rewrite !mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite transport_f_f. apply maponpaths_2. apply homset_property. - intros P₁ P₂ P₃ F G. use disp_nat_trans_eq. intros x xx. cbn ; unfold psfunctor_fib_to_psfunctor_unit_natural_data. rewrite transportf_object_cartesian_lift. rewrite !mor_disp_transportf_prewhisker. rewrite !mor_disp_transportf_postwhisker. rewrite !transport_f_f. rewrite !id_left_disp. unfold transportb. rewrite !mor_disp_transportf_prewhisker. rewrite !mor_disp_transportf_postwhisker. rewrite !transport_f_f. rewrite (disp_functor_id (pr1 G)). rewrite !id_right_disp. unfold transportb. rewrite !mor_disp_transportf_postwhisker. rewrite !transport_f_f. rewrite !id_left_disp. unfold transportb. rewrite !transport_f_f. refine (!_). rewrite assoc_disp_var. rewrite transport_f_f. rewrite assoc_disp_var. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite !mor_disp_transportf_prewhisker. rewrite transport_f_f. etrans. { do 2 apply maponpaths. rewrite assoc_disp. unfold transportb. 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. rewrite id_right_disp. unfold transportb. rewrite transport_f_f. apply idpath. } rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. rewrite cartesian_factorisation_commutes. rewrite transport_f_f. apply maponpaths_2. apply homset_property. Opaque comp_psfunctor. 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.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.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.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. 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.Morphisms.Adjunctions. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. 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.Grothendieck.FibrationToPseudoFunctor. Require Import UniMath.Bicategories.Grothendieck.PseudoFunctorToFibration.