Proposition from_left_adjoint_equivalence_fib_slice {D₁ D₂ : fib_slice_bicat} (F : D₁ --> D₂) (HF : left_adjoint_equivalence F) : is_equiv_over_id (pr1 F). Proof. simple refine (((_ ,, (_ ,, _)) ,, (_ ,, _)) ,, (_ ,, _)). - exact (pr1 (left_adjoint_right_adjoint HF)). - exact (left_adjoint_unit HF). - exact (left_adjoint_counit HF). - abstract (intros x xx ; cbn ; pose (p := maponpaths (λ z, pr1 z x xx) (internal_triangle1 HF)) ; cbn in p ; rewrite !mor_disp_transportf_postwhisker in p ; rewrite !transport_f_f in p ; rewrite id_right_disp in p ; unfold transportb in p ; rewrite transport_f_f 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 ; rewrite id_left_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 @ _) ; apply maponpaths_2 ; apply homset_property). - abstract (intros x xx ; cbn ; pose (p := maponpaths (λ z, pr1 z x xx) (internal_triangle2 HF)) ; cbn in p ; rewrite !mor_disp_transportf_postwhisker in p ; rewrite !transport_f_f in p ; rewrite id_right_disp in p ; unfold transportb in p ; rewrite transport_f_f 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 ; rewrite id_left_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 @ _) ; apply maponpaths_2 ; apply homset_property). - intros x xx. exact (from_is_invertible_2cell_fib_slice _ (pr122 HF) x xx). - intros x xx. exact (from_is_invertible_2cell_fib_slice _ (pr222 HF) x xx). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. 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.CategoryTheory.whiskering. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence.