The Stacks project

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

  1. $Y$ is integral and geometrically unibranch,

  2. $g \circ f$ is étale,

  3. 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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