 Valuative criterion for universal closedness

Among quasi-compact morphisms of schemes, universal closedness is equivalent to the existence part of the valuative criterion: for every valuation ring $D$ with fraction field $K$, every compatible pair $\mathrm{Spec}K\to X$, $\mathrm{Spec}D\to S$ extends to $\mathrm{Spec}D\to X$. All valuation rings must be tested without further hypotheses. A universally closed morphism is automatically a quasi-compact morphism.