Definition from_disp_left_adjoint_equivalence_disp_bicat_of_univ_cats {C : bicat_of_univ_cats} {D₁ D₂ : disp_bicat_of_univ_disp_cats C} (F : D₁ -->[ functor_identity _ ] D₂) (HF : disp_left_adjoint_equivalence (internal_adjoint_equivalence_identity C) F) : is_equiv_over_id F. Proof. simple refine (((_ ,, (_ ,, _)) ,, (_ ,, _)) ,, (_ ,, _)). - exact (pr11 HF). - exact (pr121 HF). - exact (pr221 HF). - abstract (intros x xx ; cbn ; pose (maponpaths (λ z, pr1 z x xx) (pr112 HF)) as p ; cbn in p ; rewrite !id_left_disp in p ; rewrite !id_right_disp in p ; unfold transportb in p ; rewrite !mor_disp_transportf_postwhisker in p ; rewrite !transport_f_f in p ; refine (transportb_transpose_right p @ _) ; etrans ; [ apply maponpaths ; apply (disp_nat_trans_transportf _ _ _ _ (!(internal_triangle1 (is_internal_adjunction_identity C))) _ _ (disp_nat_trans_id _) x xx) | ] ; unfold transportb ; rewrite transport_f_f ; apply maponpaths_2 ; apply homset_property). - abstract (intros x xx ; cbn ; pose (maponpaths (λ z, pr1 z x xx) (pr212 HF)) as p ; cbn in p ; rewrite !id_left_disp in p ; rewrite !id_right_disp in p ; unfold transportb in p ; rewrite !mor_disp_transportf_postwhisker in p ; rewrite !transport_f_f in p ; refine (transportb_transpose_right p @ _) ; etrans ; [ apply maponpaths ; apply (disp_nat_trans_transportf _ _ _ _ (!(internal_triangle2 (is_internal_adjunction_identity C))) _ _ (disp_nat_trans_id _) x xx) | ] ; unfold transportb ; rewrite transport_f_f ; apply maponpaths_2 ; apply homset_property). - abstract (cbn ; intros x xx ; apply (from_disp_bicat_of_univ_disp_cats_disp_invertible_2cell_help _ _ (pr122 HF))). - abstract (cbn ; intros x xx ; apply (from_disp_bicat_of_univ_disp_cats_disp_invertible_2cell_help _ _ (pr222 HF))). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Adjunctions. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.EquivalenceOverId. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sigma. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat.