proposition: Proposition from_left_adjoint_equivalence_opfib_slice {D₁ D₂ : opfib_slice_bicat} (F : D₁ --> D₂) (HF : left_adjoint_equivalence F) : is_equiv_over_id (pr1 F).
proof: 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_opfib_slice _ (pr122 HF) x xx). - intros x xx. exact (from_is_invertible_2cell_opfib_slice _ (pr222 HF) x xx). Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Adjunctions UniMath.CategoryTheory.DisplayedCats.Equivalences UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq UniMath.CategoryTheory.DisplayedCats.EquivalenceOverId UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq UniMath.CategoryTheory.whiskering UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.AdjointUnique UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Univalence