Lemma 29.37.18. Let $f : X \to Y$ be a morphism of schemes over $S$. If $X$ is étale over $S$ and $Y$ is unramified over $S$, then $f$ is étale.
See for example [Remark 18.30 (2)., GWII].
Proof.
Consider the factorization $X \to X \times _ S Y \to Y$, where the first arrow is given by $\text{id}_ X$ and $f$ and the second arrow is the projection. We claim both arrows are étale and hence $f$ is étale by Lemma 29.37.3. Namely, the projection is étale as it is the base change of $X \to S$, see Lemma 29.37.4. The first arrow is the base change of the diagonal morphism $Y \to Y \times _ S Y$ because the square
is cartesian. The diagonal $Y \to Y \times _ S Y$ is an open immersion by Lemma 29.36.13. The base change of an open immersion is an open immersion (Schemes, Lemma 26.18.2) and an open immersion is étale (Lemma 29.37.9).
$\square$
\[ \xymatrix{ X \ar[d] \ar[r] & X \times _ S Y \ar[d] \\ Y \ar[r] & Y \times _ S Y } \]
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