Definition from_biinitial_mor_preserves_opcartesian {x' y' : B} (f' : x' --> y') : mor_preserves_opcartesian f f' (is_biinitial_1cell_property Hx x'). Proof. intros z h₁ h₂ γ Hγ. assert(Hz : is_biinitial z). { exact (equiv_to_biinitial Hx (Sx z h₁)). } intros w ζ 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.