Definition left_equivalence_axioms_total_weq {a b : B} {aa : D a} {bb : D b} {f : a --> b} {ff : aa -->[f] bb} (td : @left_adjoint_data E (a,,aa) (b,,bb) (f,,ff)): left_equivalence_axioms td ≃ ∑ (ba : left_equivalence_axioms _), disp_left_equivalence_axioms (_,, ba) (left_adjoint_data_total_to_fiber td). Proof. apply weqimplimpl. - exact (left_equivalence_axioms_total_to_disp td). - exact (left_equivalence_axioms_disp_to_total td). - apply isapropdirprod; apply isaprop_is_invertible_2cell. - apply isofhleveltotal2. { apply isapropdirprod; apply isaprop_is_invertible_2cell. } intro. apply isapropdirprod; apply isaprop_is_disp_invertible_2cell. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.Core.Univalence.