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