Lemma 29.35.17. Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $f, g : X \to Y$ be morphisms over $S$. Let $x \in X$. Assume that
the structure morphism $Y \to S$ is unramified,
$f(x) = g(x)$ in $Y$, say $y = f(x) = g(x)$, and
the induced maps $f^\sharp , g^\sharp : \kappa (y) \to \kappa (x)$ are equal.
Then there exists an open neighbourhood of $x$ in $X$ on which $f$ and $g$ are equal.