Definition lax_slice_invertible_2cell_to_disp_adj_equiv {c : B} {t₁ t₂ : lax_slice_disp_bicat c} : invertible_2cell t₁ t₂ → disp_adjoint_equivalence (internal_adjoint_equivalence_identity c) t₁ t₂. Proof. intros α. simple refine (_ ,, ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _)))). - exact (α • linvunitor _). - exact (α^-1 • linvunitor _). - abstract (cbn ; rewrite linvunitor_natural ; rewrite <- lwhisker_hcomp ; rewrite !vassocl ; apply maponpaths ; rewrite !vassocr ; rewrite lwhisker_vcomp ; rewrite !vassocr ; rewrite vcomp_rinv ; rewrite id2_left ; rewrite lwhisker_hcomp ; rewrite triangle_l_inv ; rewrite <- rwhisker_hcomp ; apply maponpaths ; apply lunitor_V_id_is_left_unit_V_id). - abstract (cbn ; rewrite linvunitor_natural ; rewrite <- lwhisker_hcomp ; rewrite !vassocl ; refine (_ @ id2_right _) ; apply maponpaths ; rewrite !vassocr ; rewrite lwhisker_vcomp ; rewrite !vassocr ; rewrite vcomp_linv ; rewrite id2_left ; rewrite lwhisker_hcomp ; rewrite triangle_l_inv ; rewrite <- rwhisker_hcomp ; rewrite rwhisker_vcomp ; rewrite lunitor_runitor_identity ; rewrite rinvunitor_runitor ; apply id2_rwhisker). - 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.