Definition disp_map_slice_inv2cell_to_disp_adj_equiv (HB : is_univalent_2 B) {x : B} {f g : disp_map_slice_disp_bicat x} (α : invertible_2cell (pr1 f) (pr1 g)) : disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) f g. Proof. simple refine (_ ,, ((_ ,, (_ ,, _)) ,, (_ ,, _))). - simple refine (_ ,, _). + apply (arrow_subbicat_contains_equiv_over_id HB). * apply internal_adjoint_equivalence_identity. * exact (comp_of_invertible_2cell α (linvunitor_invertible_2cell _)). + exact (comp_of_invertible_2cell α (linvunitor_invertible_2cell _)). - simple refine (_ ,, _). + apply (arrow_subbicat_contains_equiv_over_id HB). * apply internal_adjoint_equivalence_identity. * exact (comp_of_invertible_2cell (inv_of_invertible_2cell α) (linvunitor_invertible_2cell _)). + exact (comp_of_invertible_2cell (inv_of_invertible_2cell α) (linvunitor_invertible_2cell _)). - 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 ; rewrite lunitor_V_id_is_left_unit_V_id ; apply idpath). - 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 !vassocl ; use vcomp_move_R_pM ; [ is_iso | ] ; cbn ; rewrite id2_right ; rewrite lunitor_runitor_identity ; rewrite runitor_rwhisker ; apply idpath). - abstract (split ; apply disp_2cells_isaprop_disp_map_slice). - abstract (split ; apply disp_locally_groupoid_disp_map_slice_disp_bicat). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Discreteness. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Logic.DisplayMapBicat. Require Import UniMath.Bicategories.Limits.Pullbacks.