Definition 59.35.1. Let $f: X\to Y$ be a morphism of schemes. Let $\mathcal{F}$ a presheaf of sets on $X_{\acute{e}tale}$. The direct image, or pushforward of $\mathcal{F}$ (under $f$) is

$f_*\mathcal{F} : Y_{\acute{e}tale}^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times _ Y V/X).$

We sometimes write $f_* = f_{small, *}$ to distinguish from other direct image functors (such as usual Zariski pushforward or $f_{big, *}$).

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