Definition add_cell_disp_cat_univalent_2_0 (HC : is_univalent_2_1 C) (HD : disp_univalent_2_1 D) : disp_univalent_2_0 add_cell_disp_cat. Proof. use disp_cell_unit_bicat_univalent_2_0. - apply total_is_univalent_2_1. + exact HC. + exact HD. - intros. apply C. - intros x xx yy. simpl in *. apply C. - abstract (intros x xx yy; intros p; induction p as [p q]; cbn; cbn in p, q; pose (pstrans_id_alt l) as pl; cbn in pl ; rewrite pl in p ; clear pl; pose (pstrans_id_alt r) as pr; cbn in pr ; rewrite pr in p ; clear pr; cbn in p; rewrite !psfunctor_id2 in p; rewrite !id2_right, !id2_left in p; rewrite !vassocr in p; rewrite vcomp_whisker in p; rewrite !vassocl in p; assert (is_invertible_2cell (l x ◃ ((pr122 T) (pr1 x)) ^-1)) as H; try is_iso ; pose (vcomp_lcancel _ H p) as p'; rewrite !vassocr in p'; rewrite vcomp_runitor in p'; rewrite !vassocl in p'; pose (vcomp_lcancel _ (is_invertible_2cell_runitor _) p') as p''; use (vcomp_rcancel (linvunitor (r x))) ; try is_iso; use (vcomp_rcancel (psfunctor_id S (pr1 x) ▹ r x)) ; try (is_iso ; exact (psfunctor_id S (pr1 x))); rewrite !vassocl; refine (p'' @ _); rewrite vcomp_whisker; rewrite !vassocr; apply maponpaths_2; rewrite lwhisker_hcomp; exact (!(linvunitor_natural _))). 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.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Examples.OneTypes. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Projection.