Definition dfl_full_comp_cat_left_adjoint_equivalence_from_full_comp_cat {C₁ C₂ : dfl_full_comp_cat} (F : dfl_full_comp_cat_functor C₁ C₂) (HF : left_adjoint_equivalence (pr1 F)) : left_adjoint_equivalence F. Proof. use (invmap (left_adjoint_equivalence_total_disp_weq _ _)). refine (HF ,, _). use (pair_left_adjoint_equivalence _ _ (_ ,, HF)). split. - apply disp_left_adjoint_equivalence_fullsubbicat. - use (pair_left_adjoint_equivalence _ _ (_ ,, HF)). split. + apply disp_adjoint_equiv_disp_bicat_of_strong_unit_type. + use (pair_left_adjoint_equivalence _ _ (_ ,, HF)). split. * apply disp_adjoint_equiv_disp_bicat_of_prod_type_full_comp_cat. * use (pair_left_adjoint_equivalence _ _ (_ ,, HF)). split. ** apply disp_adjoint_equiv_disp_bicat_of_equalizer_type_full_comp_cat. ** apply disp_adjoint_equiv_disp_bicat_of_sigma_type_full_comp_cat. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseTerminal. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseEqualizers. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DependentSums. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.PreservationProperties. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.StructuredCategories. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat. Require Import UniMath.Bicategories.ComprehensionCat.CompCatNotations.