Definition lax_slice_disp_adj_equiv_to_invertible_2cell {c : B} {t₁ t₂ : lax_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 (pr1 e • lunitor _). - use make_is_invertible_2cell. + exact (pr112 e • lunitor _). + abstract (pose (pr1 (pr212 e)) as m ; cbn in m ; rewrite !vassocl in m ; rewrite !vassocr ; use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ; rewrite id2_left ; rewrite !vassocl ; rewrite <- vcomp_lunitor ; rewrite <- lunitor_triangle ; rewrite !vassocr ; use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ; rewrite !vassocl ; exact (!m)). + abstract (pose (pr2 (pr212 e)) as m ; cbn in m ; rewrite !vassocr ; use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ; rewrite id2_left ; rewrite !vassocl ; rewrite <- vcomp_lunitor ; rewrite <- lunitor_triangle ; rewrite !vassocr ; exact m). 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.