Lemma 54.6.3. 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 $X_{s_ i} \cong (Y_ i)_{s_ i}$ as schemes over $\kappa (s_ i)$.

**Proof.**
This is clear.
$\square$

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