Proposition psfunctor_fib_to_psfunctor_counit_data_laws_on_ob (F : psfunctor (cat_to_bicat C^op) bicat_of_univ_cats) : pstrans_from_cat_into_cat_laws (psfunctor_fib_to_psfunctor_counit_data_data_on_ob F). Proof. split. - intros x xx ; cbn -[psfunctor_id]. unfold psfunctor_fib_to_psfunctor_counit_data_ob_nat_trans_data. refine (id_right _ @ _). refine (!_). etrans. { apply maponpaths_2. cbn. unfold is_cartesian_indexed_cat_factorisation. cbn -[psfunctor_id psfunctor_comp]. apply idpath. } rewrite functor_id. rewrite id_right. refine (assoc' _ _ _ @ _). etrans. { apply maponpaths. etrans. { pose (p := nat_trans_eq_pointwise (psfunctor_F_runitor F (identity x)) xx). cbn -[psfunctor_id psfunctor_comp] in p. refine (_ @ !p). rewrite id_right. apply idpath. } apply psfunctor_idtoiso. } etrans. { apply maponpaths_2. apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat F)). } refine (assoc' _ _ _ @ _ @ id_right _). apply maponpaths. refine (!(pr1_idtoiso_concat _ _) @ _ @ idtoiso_idpath _). do 2 apply maponpaths. refine (!(maponpathscomp0 (λ z, pr1 (#(pr11 F) z) _) _ _) @ _). refine (_ @ @maponpaths_idpath _ _ (λ z, pr1 (#(pr11 F) z) _) _). apply maponpaths. apply homset_property. - intros x y z f g xx ; cbn -[psfunctor_id psfunctor_comp]. unfold psfunctor_fib_to_psfunctor_counit_data_ob_nat_trans_data. rewrite !id_right. refine (!_). etrans. { apply maponpaths_2. apply functor_id. } rewrite id_left. etrans. { apply maponpaths_2. cbn. unfold is_cartesian_indexed_cat_factorisation. cbn -[psfunctor_comp]. rewrite !functor_id. rewrite !id_right. apply maponpaths_2. apply maponpaths. apply id_left. } refine (assoc' _ _ _ @ _). etrans. { apply maponpaths_2. apply (transportf_indexed_cat_to_disp_cat (psfunctor_to_indexed_cat F)). } refine (assoc' _ _ _ @ _ @ id_right _). apply maponpaths. etrans. { apply maponpaths. cbn in g, f. pose (p := nat_trans_eq_pointwise (psfunctor_F_runitor F (g · f)) xx). cbn -[psfunctor_id psfunctor_comp] in p. refine (_ @ !p). rewrite id_right. apply idpath. } etrans. { apply maponpaths. apply psfunctor_idtoiso. } refine (!(pr1_idtoiso_concat _ _) @ _ @ idtoiso_idpath _). do 2 apply maponpaths. refine (!(maponpathscomp0 (λ z, pr1 (#(pr11 F) z) _) _ _) @ _). refine (_ @ @maponpaths_idpath _ _ (λ z, pr1 (#(pr11 F) z) _) _). apply maponpaths. 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.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.