Lemma 35.11.4. Let

be a commutative diagram of morphisms of schemes. Assume that

$f$ is surjective, and syntomic (resp. smooth, resp. étale),

$p$ is syntomic (resp. smooth, resp. étale).

Then $q$ is syntomic (resp. smooth, resp. étale).

Lemma 35.11.4. 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 syntomic (resp. smooth, resp. étale),

$p$ is syntomic (resp. smooth, resp. étale).

Then $q$ is syntomic (resp. smooth, resp. étale).

**Proof.**
Combine Morphisms, Lemmas 29.30.16, 29.34.19, and 29.36.19 with Lemma 35.11.3 above.
$\square$

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