Definition weq_disp_left_adjequiv_hor_comp {CD : bicat_twosided_disp_cat} (FF GG : disp_bicat_twosided_disp_cat_hor_comp CD) : (∑ (F : ∏ (x y z : pr11 CD) (h : pr12 CD x y) (k : pr12 CD y z), iso_twosided_disp (identity_z_iso _) (identity_z_iso _) (double_hor_comp (pr1 GG) h k) (double_hor_comp (pr1 FF) h k)), double_functor_hor_comp_laws (FF := twosided_disp_functor_identity _) (Cm₁ := FF) (Cm₂ := GG) (λ x y z h k, pr1 (F x y z h k))) ≃ disp_adjoint_equivalence (internal_adjoint_equivalence_identity CD) FF GG. Proof. use weq_iso. - intros F. exact (weq_disp_left_adjequiv_hor_comp_map (pr1 F) (pr2 F)). - exact weq_disp_left_adjequiv_hor_comp_inv. - abstract (intro τ ; use subtypePath ; [ intro ; apply isaprop_double_functor_hor_comp_laws | ] ; repeat (use funextsec ; intro) ; use subtypePath ; [ intro ; apply isaprop_is_iso_twosided_disp | ] ; cbn ; apply idpath). - abstract (intro τ ; use subtypePath ; [ intro ; apply isaprop_disp_left_adjoint_equivalence ; [ apply is_univalent_2_1_bicat_twosided_disp_cat | apply disp_univalent_2_1_disp_bicat_twosided_disp_cat_hor_comp ] | ] ; use subtypePath ; [ intro ; apply isaprop_double_functor_hor_comp_laws | ] ; use funextsec ; intro ; cbn ; apply idpath). Defined.
Require Import UniMath.MoreFoundations.All. 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.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DispBicatOfDispCats. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DispBicatOfTwoSidedDispCat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleTransformation. Section FromAdjEquivHorId. End FromAdjEquivHorId. Section FromAdjEquivHorComp. End FromAdjEquivHorComp.