= Solution
With the same compatible pair $u,v$, the <valuative criterion for properness> is
$$
\boxed{f\text{ proper}\iff f\text{ is of finite type and quasi-separated, and every pair has exactly one extension}.}
$$
Equivalently, one can replace <finite type> by being a <quasi-compact morphism> and <locally of finite type>. Necessity follows from the definition of a <proper morphism> and the existence and uniqueness criteria in (i) and (ii). Conversely, <finite type> supplies the compactness needed in (i), while being a <quasi-separated morphism> supplies the compactness of the <diagonal morphism> needed in (ii). Thus the extension condition gives both <universal closedness> and <separatedness>, and these together with <finite type> give a <proper morphism>. These hypotheses are part of the criteria, even though the question is phrased for unrestricted <morphisms of schemes>.https://stacks.math.columbia.edu/tag/0BX5[Stacks Project, valuative criterion for properness]{ref}
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