Proposition 41.6.3. Let $S$ is be a scheme. Let $\pi : X \to S$ be unramified and separated. Let $Y$ be an $S$-scheme and $y \in Y$ a point. Let $f, g : Y \to X$ be two $S$-morphisms. Assume
$Y$ is connected
$x = f(y) = g(y)$, and
the induced maps $f^\sharp , g^\sharp : \kappa (x) \to \kappa (y)$ on residue fields are equal.
Then $f = g$.