Definition from_biinitial_is_cartesian_2cell_sfib {z : B} {g₁ g₂ : z --> x} (α : g₁ ==> g₂) : is_cartesian_2cell_sfib f α. Proof. assert(Hz : is_biinitial z). { exact (equiv_to_biinitial Hx (Sx z g₁)). } intros w γ δp p. use iscontraprop1. - abstract (use invproofirrelevance ; intros φ₁ φ₂ ; use subtypePath ; [ intro ; apply isapropdirprod ; apply cellset_property | ] ; apply (is_biinitial_eq_property Hz)). - simple refine (_ ,, _ ,, _). + apply (is_biinitial_2cell_property Hz). + apply (is_biinitial_eq_property Hz). + apply (is_biinitial_eq_property Hz). 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.Examples.OpCellBicat. 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.Colimits.Initial. Section FromInitial. End FromInitial.