∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) (X₀ : C)
  (f : X.obj (Opposite.op (SimplexCategory.mk 0)) ⟶ X₀)
  (w :
    ∀ (i : SimplexCategory) (g₁ g₂ : SimplexCategory.mk 0 ⟶ i),
      CategoryTheory.CategoryStruct.comp (X.map g₁.op) f = CategoryTheory.CategoryStruct.comp (X.map g₂.op) f),
  (X.augment X₀ f w).hom.app (Opposite.op (SimplexCategory.mk 0)) = f