The Stacks project

Remark 59.44.2. Property (C) holds if $f : X \to Y$ is an open immersion. Namely, if $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$, then we can view $U$ also as an object of $Y_{\acute{e}tale}$ and $U \times _ Y X = U$. Hence property (C) does not imply that $f_{small, *}$ is exact as this is not the case for open immersions (in general).