Definition arrow_subbicat_contains_equiv_over_id {B : bicat} (HB : is_univalent_2 B) (D : arrow_subbicat B) : contains_equiv_over_id (λ e₁ e₂ b₁ b₂ p₁ p₂ fe, pred_mor D p₁ p₂ fe). Proof. intros e₁ e₂ b p₁ p₂ fe Hfe γ. refine (J_2_0 (pr1 HB) (λ (x₁ x₂ : B) (L : adjoint_equivalence x₁ x₂), ∏ (p₁ : x₁ --> b) (p₂ : x₂ --> b) (γ : invertible_2cell p₁ (pr1 L · p₂)), pred_mor D p₁ p₂ (pr1 L)) _ (fe ,, Hfe) p₁ p₂ γ). cbn. clear e₁ e₂ p₁ p₂ fe Hfe γ. intros e p₁ p₂ γ. pose (c := comp_of_invertible_2cell γ (lunitor_invertible_2cell _)). use (J_2_1 (pr2 HB) (λ (x₁ x₂ : B) (f g : x₁ --> x₂) (γ : invertible_2cell f g), pred_mor D f g (id₁ _)) _ c). cbn ; intros. apply (id_pred_mor D). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Morphisms.Properties.ContainsAdjEquiv. Require Import UniMath.Bicategories.Morphisms.Properties.Composition. Require Import UniMath.Bicategories.Morphisms.Properties.ClosedUnderPullback. Require Import UniMath.Bicategories.Morphisms.Properties.FromInitial. Require Import UniMath.Bicategories.Colimits.Initial. Require Import UniMath.Bicategories.Limits.Pullbacks.