Definition oplax_slice_invertible_2cell_to_disp_adj_equiv {c : B} {t₁ t₂ : oplax_slice_disp_bicat c} : invertible_2cell t₁ t₂ → disp_adjoint_equivalence (internal_adjoint_equivalence_identity c) t₁ t₂. Proof. intros α. simple refine (_ ,, ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _)))). - exact (lunitor _ • α^-1). - exact (lunitor _ • α). - abstract (cbn ; rewrite !vassocr ; rewrite <- linvunitor_assoc ; rewrite lwhisker_hcomp ; rewrite <- linvunitor_natural ; rewrite !vassocl ; refine (_ @ id2_right _) ; apply maponpaths ; rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ; rewrite linvunitor_lunitor ; rewrite id2_left ; apply vcomp_rinv). - abstract (cbn ; rewrite <- !lwhisker_vcomp ; rewrite !vassocr ; rewrite lunitor_lwhisker ; rewrite runitor_lunitor_identity ; rewrite !vassocl ; apply maponpaths ; rewrite !vassocr ; rewrite vcomp_lunitor ; rewrite !vassocl ; rewrite vcomp_linv ; rewrite id2_right ; apply idpath). - apply oplax_slice_disp_2cells_isaprop. - apply oplax_slice_disp_2cells_isaprop. - apply oplax_slice_disp_locally_groupoid. - apply oplax_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.