Lemma 29.36.19. Let

be a commutative diagram of morphisms of schemes. Assume that

$f$ is surjective, and étale,

$p$ is étale, and

$q$ is locally of finite presentation

^{1}.

Then $q$ is étale.

Lemma 29.36.19. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that

$f$ is surjective, and étale,

$p$ is étale, and

$q$ is locally of finite presentation

^{1}.

Then $q$ is étale.

**Proof.**
By Lemma 29.34.19 we see that $q$ is smooth. Thus we only need to see that $q$ has relative dimension $0$. This follows from Lemma 29.28.2 and the fact that $f$ and $p$ have relative dimension $0$.
$\square$

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