Definition cleaving_of_cleaving_lift_mor_lift_1cell {C₁ C₂ C₃ : bicat_of_univ_cats} {D₂ : disp_bicat_of_cleaving C₂} {D₃ : disp_bicat_of_cleaving C₃} {F : C₁ --> C₂} {H : C₃ --> C₁} (HH : D₃ -->[ H · F] D₂) : lift_1cell_factor disp_bicat_of_cleaving (cleaving_of_cleaving_lift_mor D₂ F) HH. Proof. simple refine (_ ,, _). - simple refine (_ ,, _). + exact (lift_functor_into_reindex (pr1 HH)). + exact (is_cartesian_lift_functor_into_reindex (pr2 HH)). - simple refine ((_ ,, tt) ,, _). + exact (lift_functor_into_reindex_commute (pr1 HH)). + apply disp_bicat_of_cleaving_is_disp_invertible_2cell. intros x xx. apply id_is_z_iso_disp. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Reindexing. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DispBicatOfDispCats.