Definition oplax_slice_disp_adj_equiv_to_invertible_2cell {c : B} {t₁ t₂ : oplax_slice_disp_bicat c} : disp_adjoint_equivalence (internal_adjoint_equivalence_identity c) t₁ t₂ → invertible_2cell t₁ t₂. Proof. intros e. use make_invertible_2cell. - exact (linvunitor _ • pr112 e). - use make_is_invertible_2cell. + exact (linvunitor _ • pr1 e). + abstract (rewrite !vassocl ; use vcomp_move_R_pM ; [ is_iso | ] ; cbn ; rewrite id2_right ; refine (_ @ pr1 (pr212 e)) ; cbn ; rewrite !vassocr ; apply maponpaths_2 ; rewrite linvunitor_natural ; rewrite <- lwhisker_hcomp ; apply maponpaths_2 ; rewrite linvunitor_assoc ; apply idpath). + abstract (rewrite !vassocl ; use vcomp_move_R_pM ; [ is_iso | ] ; cbn ; rewrite id2_right ; rewrite !vassocr ; rewrite linvunitor_natural ; rewrite <- lwhisker_hcomp ; rewrite linvunitor_assoc ; rewrite !vassocl ; use vcomp_move_R_pM ; [ is_iso | ] ; cbn ; rewrite !vassocr ; exact (!(pr2 (pr212 e)))). 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.