The Stacks project

Remark 59.35.2. We claim that the direct image of a sheaf is a sheaf. Namely, if $\{ V_ j \to V\} $ is an étale covering in $Y_{\acute{e}tale}$ then $\{ X \times _ Y V_ j \to X \times _ Y V\} $ is an étale covering in $X_{\acute{e}tale}$. Hence the sheaf condition for $\mathcal{F}$ with respect to $\{ X \times _ Y V_ i \to X \times _ Y V\} $ is equivalent to the sheaf condition for $f_*\mathcal{F}$ with respect to $\{ V_ i \to V\} $. Thus if $\mathcal{F}$ is a sheaf, so is $f_*\mathcal{F}$.