Definition cleaving_of_opcleaving_lift_mor_lift_2cell : lift_2cell_factor _ _ αα Lh Lh'. Proof. use iscontraprop1. - use invproofirrelevance. intros φ₁ φ₂. exact (cleaving_of_opcleaving_unique_2_lifts φ₁ φ₂). - simple refine ((_ ,, tt) ,, _). + exact cleaving_of_opcleaving_lift_2cell. + abstract (cbn ; use subtypePath ; [ intro ; apply isapropunit | ] ; use disp_nat_trans_eq ; intros x xx ; cbn ; rewrite pr1_transportf ; unfold disp_cell_lift_1cell_factor ; refine (@disp_nat_trans_transportf _ _ _ _ (H₁ ∙ F) (H₂ ∙ F) _ _ (id2_right (α ▹ F) @ ! id2_left (α ▹ F)) (disp_functor_composite (pr11 Lh) (reindex_disp_cat_disp_functor F (pr11 D₂))) (pr1 HH₂) (disp_nat_trans_comp (post_whisker_disp_nat_trans cleaving_of_opcleaving_lift_2cell (reindex_disp_cat_disp_functor F (pr11 D₂))) (pr112 Lh')) x xx @ _) ; cbn ; unfold cleaving_of_opcleaving_lift_2cell_data ; rewrite !mor_disp_transportf_postwhisker ; rewrite !transport_f_f ; rewrite !assoc_disp_var ; rewrite !transport_f_f ; etrans ; [ do 3 apply maponpaths ; apply (z_iso_disp_after_inv_mor (disp_bicat_of_opcleaving_disp_invertible_2cell_pointwise_inv (id2_invertible_2cell (H₂ · F)) (pr2 Lh') (pr22 Lh') xx)) | ] ; unfold transportb ; rewrite !mor_disp_transportf_prewhisker ; rewrite transport_f_f ; rewrite id_right_disp ; unfold transportb ; rewrite mor_disp_transportf_prewhisker ; rewrite transport_f_f ; apply transportf_set ; apply homset_property). 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.CategoryTheory.DisplayedCats.Examples.Opposite. 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.