Lemma 54.6.2. Let $S, s_ i, S_ i$ be as in (54.6.0.1). If $f : X \to S$ corresponds to $g_ i : Y_ i \to S_ i$ under $F$, then $f$ is separated, proper, finite, if and only if $g_ i$ is so for $i = 1, \ldots , n$.

**Proof.**
Follows from Limits, Lemma 32.21.2.
$\square$

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