Definition 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 : is_equiv_over_id F) : disp_left_adjoint_equivalence (internal_adjoint_equivalence_identity C) F. Proof. simple refine ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _))). - exact (pr111 HF). - exact (pr1 (pr211 HF)). - exact (pr2 (pr211 HF)). - abstract (use disp_nat_trans_eq ; intros x xx ; cbn ; rewrite id_left_disp ; rewrite !id_right_disp ; unfold transportb ; rewrite !mor_disp_transportf_postwhisker ; rewrite !transport_f_f ; etrans ; [ apply maponpaths ; exact (pr121 HF x xx) | ] ; unfold transportb ; rewrite transport_f_f ; refine (!_) ; refine (disp_nat_trans_transportf _ _ _ _ (!(internal_triangle1 (is_internal_adjunction_identity C))) _ _ (disp_nat_trans_id _) x xx @ _) ; apply maponpaths_2 ; apply homset_property). - abstract (use disp_nat_trans_eq ; intros x xx ; cbn ; rewrite id_left_disp ; rewrite !id_right_disp ; unfold transportb ; rewrite !mor_disp_transportf_postwhisker ; rewrite !transport_f_f ; etrans ; [ apply maponpaths ; exact (pr221 HF x xx) | ] ; unfold transportb ; rewrite transport_f_f ; refine (!_) ; refine (disp_nat_trans_transportf _ _ _ _ (!(internal_triangle2 (is_internal_adjunction_identity C))) _ _ (disp_nat_trans_id _) x _ @ _) ; apply maponpaths_2 ; apply homset_property). - abstract (use disp_bicat_of_univ_disp_cats_is_disp_invertible_2cell_help ; cbn ; intros x xx ; exact (pr12 HF x xx)). - abstract (use disp_bicat_of_univ_disp_cats_is_disp_invertible_2cell_help ; cbn ; intros x xx ; exact (pr22 HF x xx)). 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.