proposition: Proposition fiberwise_univalent_2_0_disp_bicat_comp_cat : fiberwise_univalent_2_0 disp_bicat_comp_cat.
proof: Proof. refine (λ (C : cat_with_terminal_cleaving) (χ₁ χ₂ : comprehension_functor C), _). use weqhomot. - exact (disp_nat_z_iso_weq_disp_adjequiv χ₁ χ₂ ∘ disp_functor_eq_weq χ₁ χ₂ (disp_univalent_disp_codomain C) ∘ path_sigma_hprop _ _ _ (isaprop_is_cartesian_disp_functor _))%weq. - intro p ; cbn in p. induction p. use subtypePath. { intro. use (isaprop_disp_left_adjoint_equivalence _ _ is_univalent_2_1_bicat_cat_with_terminal_cleaving). exact disp_univalent_2_1_disp_bicat_comp_cat. } use disp_nat_trans_eq. intros x xx. use subtypePath. { intro. apply homset_property. } apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq UniMath.CategoryTheory.DisplayedCats.ComprehensionC UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Preservation UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispAdjunctions UniMath.Bicategories.DisplayedBicats.DispInvertibles UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat