Lemma 59.104.8. Let $S$ be a scheme. Then the category fibred in groupoids
whose fibre category over $U$ is the category $\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$ of sheaves on the small étale site of $U$ is a stack in groupoids.
Proof. To prove the lemma we will check conditions (1), (2), and (3) of More on Flatness, Lemma 38.37.13.
Condition (1) holds because we have glueing for sheaves (and Zariski coverings are étale coverings). See Sites, Lemma 7.26.4.
To see condition (2), suppose that $f : X \to Y$ is a surjective, flat, proper morphism of finite presentation over $S$ with $Y$ affine. Then we have descent for $\{ X \to Y\} $ by either Lemma 59.104.5 or Lemma 59.104.3.
Condition (3) follows immediately from the more general Lemma 59.104.7. $\square$
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