Lemma 37.37.4 (0GSB)—The Stacks project

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Lemma 37.37.4. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. Assume

$Y$ is integral and geometrically unibranch,
$g \circ f$ is étale,
every irreducible component of $X$ dominates $Y$ .

Then $f$ is étale and $g$ is étale at every point in the image of $f$ .

Proof. Immediate from the pointwise version Lemma 37.37.3. $\square$

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