proposition: Lemma adjunction_total_disp_weq {a b : B} (aa : D a) (bb : D b) : @adjunction E (a,,aa) (b,,bb) ≃ ∑ (f : adjunction a b), disp_adjunction f aa bb.
proof: Proof. unfold adjunction. cbn. eapply weqcomp. { apply weqinvweq. apply weqtotal2asstol. } eapply weqcomp. 2: { apply weqtotal2asstol. } eapply weqfibtototal. intros f. unfold disp_adjunction. cbn. eapply weqcomp. 2: { apply weqtotal2comm. } eapply weqfibtototal. intros ff. unfold left_adjoint. eapply weqcomp. { pose (Q:=λ (w : (∑ αd, disp_left_adjoint_data αd ff)), (∑ l, disp_left_adjoint_axioms (pr1 w,,l) (pr2 w))). eapply (weqbandf (left_adjoint_data_total_weq ff) _ Q). intros td. unfold Q; cbn. apply left_adjoint_axioms_total_weq. } cbn. eapply weqcomp. { apply weqinvweq. apply weqtotal2asstol. } cbn. eapply weqcomp. 2: { apply weqtotal2asstol. } cbn. eapply weqfibtototal. intros αd. eapply weqcomp. { apply weqtotal2comm. } cbn. apply idweq. Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.Bicategories.Core.Bicat UniMath.CategoryTheory.DisplayedCats.Core UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.AdjointUnique UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.DisplayedBicats.DispInvertibles UniMath.Bicategories.Core.Univalence