V be the effective inclusion. In the following we assume C has enough valuation objects. Then the restrictions j , j ' on the generic point K of V are equal. These two morphisms must be equal. This shows that g factors through D f .
V is a stable morphism and V is a valuation object.
V as any point of V is a specialization of K . X is a section of f making the diagram commutative. Let y ' be a specialization of y . X making that diagram commute. V is a stable morphism.