= Solution
Use the compatible pair $u,v$ and the notion of extension from (i). The <valuative criterion for separatedness> is
$$
\boxed{f\text{ separated}\iff f\text{ quasi-separated and every such pair has at most one extension}.}
$$
A <quasi-separated morphism> means one whose <diagonal morphism> is a <quasi-compact morphism>. A <separated morphism> automatically has this property because its <diagonal morphism> is a <closed immersion>. The <quasi-separated morphism> hypothesis is needed in the converse valuative test for arbitrary <schemes>. Existence is not demanded.https://stacks.math.columbia.edu/tag/01KY[Stacks Project, valuative criterion of separatedness]{ref}
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