Definition lax_slice_invertible_2cell_is_left_disp_adj_equiv {c : B} {t₁ t₂ : lax_slice_disp_bicat c} (f : t₁ -->[ id₁ _ ] t₂) (Hf : is_invertible_2cell f) : disp_left_adjoint_equivalence (internal_adjoint_equivalence_identity c) f. Proof. simple refine ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _))). - exact (linvunitor _ • Hf^-1 • linvunitor _). - abstract (cbn ; rewrite <- !lwhisker_vcomp ; rewrite !vassocl ; rewrite !lwhisker_hcomp ; rewrite triangle_l_inv ; rewrite <- !lwhisker_hcomp, <- rwhisker_hcomp ; rewrite lunitor_V_id_is_left_unit_V_id ; rewrite !vassocr ; apply maponpaths_2 ; use vcomp_move_L_Mp ; [ is_iso | ] ; cbn ; rewrite lwhisker_hcomp ; rewrite <- linvunitor_natural ; apply maponpaths ; rewrite linvunitor_assoc ; rewrite lunitor_V_id_is_left_unit_V_id ; rewrite rwhisker_hcomp ; rewrite <- triangle_l_inv ; rewrite <- lwhisker_hcomp ; rewrite !vassocl ; rewrite lassociator_rassociator ; apply id2_right). - abstract (cbn ; rewrite !vassocl ; refine (_ @ id2_right _) ; apply maponpaths ; do 2 (use vcomp_move_R_pM ; [ is_iso | ]) ; cbn ; rewrite id2_right ; rewrite <- vcomp_lunitor ; rewrite lunitor_triangle ; apply idpath). - apply lax_slice_disp_2cells_isaprop. - apply lax_slice_disp_2cells_isaprop. - apply lax_slice_disp_locally_groupoid. - apply lax_slice_disp_locally_groupoid. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.