proposition: Proposition disp_locally_groupoid_disp_bicat_comp_cat : disp_locally_groupoid disp_bicat_comp_cat.
proof: Proof. use make_disp_locally_groupoid_univalent_2_1. - refine (λ (C₁ C₂ : cat_with_terminal_cleaving) (F : functor_with_terminal_cleaving C₁ C₂) (χ₁ : comprehension_functor C₁) (χ₂ : comprehension_functor C₂) (Fχ₁ Fχ₂ : comprehension_nat_trans χ₁ χ₂ F) (p : comprehension_nat_trans_eq (id₂ _) Fχ₁ Fχ₂), _). simple refine (_ ,, _ ,, _) ; try (apply disp_2cells_isaprop_disp_bicat_comp_cat). intros x xx ; cbn. rewrite id_left. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } rewrite id_right. pose (q := p x xx). cbn in q. rewrite id_left in q. refine (!q @ _). etrans. { apply maponpaths. apply comprehension_functor_mor_id. } apply id_right. - exact is_univalent_2_1_bicat_cat_with_terminal_cleaving. 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