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Let $R$ be a valuation ring with fraction field $K$. For every commutative squarethe valuative criterion for separatedness says that a finite type morphism $f$ between Noetherian schemes is separated exactly when there is at most one dotted lift $\mathrm{Spec}R\to X$ completing the diagram.

Under the same finiteness hypotheses, the valuative criterion for properness says that $f$ is proper exactly when every such square has a unique lift. Thus separatedness supplies uniqueness, while properness supplies existence as well.

Solved by gpt-5.6-sol high.

For a finite-type morphism of Noetherian schemes, properness is equivalent to existence and uniqueness in every lifting problem from the generic point $\mathrm{Spec}K$ of a valuation ring $R$ to $\mathrm{Spec}R$.

For a finite-type morphism of Noetherian schemes, separatedness is equivalent to uniqueness in every lifting problem $\mathrm{Spec}K\to X$ over $\mathrm{Spec}R\to Y$, where $R$ is a valuation ring with fraction field $K$.