Definition cat_with_terminal_cleaving_left_adjoint_equivalence {C₁ C₂ : cat_with_terminal_cleaving} (F : functor_with_terminal_cleaving C₁ C₂) (HF₁ : adj_equivalence_of_cats F) (A := (_ ,, HF₁) : adj_equiv C₁ C₂) (HF₂ : is_equiv_over A (comp_cat_type_functor F)) : left_adjoint_equivalence F. Proof. use left_adjoint_equivalence_subbicat. - clear C₁ C₂ F HF₁ A HF₂. intros C₁ C₂ HC₁ HC₂ F HF. exact (cat_with_terminal_cleaving_left_adjoint_equivalence_help C₁ C₂ (F ,, HF)). - use cat_with_terminal_disp_cat_left_adjoint_equivalence. + exact HF₁. + exact HF₂. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. 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.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell.