Definition disp_depprod_bicat_disp_adjequiv_inv {a : B} (aa bb : disp_depprod_bicat a) : disp_adjoint_equivalence (internal_adjoint_equivalence_identity a) aa bb → (∏ (i : I), disp_adjoint_equivalence (internal_adjoint_equivalence_identity a) (aa i) (bb i)). Proof. intros f i. use tpair. - exact (pr1 f i). - use tpair. + use tpair. * exact (disp_left_adjoint_right_adjoint _ f i). * split. ** exact (disp_left_adjoint_unit _ f i). ** exact (disp_left_adjoint_counit _ f i). + split. * split. ** abstract (refine (eqtohomot (pr1 (pr122 f)) i @ _) ; refine (!(disp_depprod_prebicat_laws_help _ _ _))). ** abstract (refine (eqtohomot (pr2 (pr122 f)) i @ _) ; refine (!(disp_depprod_prebicat_laws_help _ _ _))). * split. ** exact (disp_depprod_bicat_disp_is_invertible_2cell_pr _ (pr1 (pr222 f)) i). ** exact (disp_depprod_bicat_disp_is_invertible_2cell_pr _ (pr2 (pr222 f)) i). 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.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence.