Lemma 89.3.2. Let $X, x_ i, U_ i \to X, u_ i$ be as in (89.3.0.1). If $f : Y \to X$ corresponds to $g_ i : Y_ i \to U_ i$ under $F$, then $f$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, if and only if $g_ i$ is so for $i = 1, \ldots , n$.
Proof. Follows from Limits of Spaces, Lemma 70.19.2. $\square$
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