The Stacks project

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

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