Definition disp_alg_bicat_adjoint_equivalence_weq (HC : is_univalent_2_1 C) {a : C} (aa bb : disp_alg_bicat a) : invertible_2cell aa bb ≃ disp_adjoint_equivalence (internal_adjoint_equivalence_identity a) aa bb. Proof. use make_weq. - exact (disp_alg_bicat_adjoint_equivalence aa bb). - use isweq_iso. + exact (disp_alg_bicat_adjoint_equivalence_inv aa bb). + intros x. use subtypePath. * intro. apply isaprop_is_invertible_2cell. * cbn. abstract (rewrite invertible_2cell_map_alg_eq2 ; unfold invertible_2cell_map_alg'' ; cbn ; unfold left_adjoint_2cell ; rewrite !vassocr ; rewrite rinvunitor_runitor, id2_left ; rewrite !vassocl ; rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)) ; rewrite rwhisker_vcomp, vcomp_rinv ; rewrite id2_rwhisker, id2_left ; rewrite linvunitor_lunitor, id2_right ; reflexivity). + intros x. use subtypePath. * intro. apply isaprop_disp_left_adjoint_equivalence. ** exact HC. ** exact disp_alg_bicat_univalent_2_1. * cbn. unfold left_adjoint_2cell. unfold disp_alg_bicat_adjoint_equivalence_inv ; cbn. apply subtypePath. { intro ; apply isaprop_is_invertible_2cell. } cbn. abstract (rewrite invertible_2cell_map_alg_eq2 ; unfold invertible_2cell_map_alg'' ; cbn ; unfold left_adjoint_2cell ; rewrite !vassocr ; rewrite runitor_rinvunitor, id2_left ; rewrite !vassocl ; rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)) ; rewrite lunitor_linvunitor, id2_left ; rewrite rwhisker_vcomp, vcomp_linv ; rewrite id2_rwhisker, id2_right ; reflexivity). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.