Lemma 59.100.1. With notation as above. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. The rule

is a sheaf and a fortiori a sheaf on $(\mathit{Sch}/S)_{\acute{e}tale}$. In fact this sheaf is equal to $\pi _ S^{-1}\mathcal{F}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$ and $\epsilon _ S^{-1}\pi _ S^{-1}\mathcal{F}$ on $(\mathit{Sch}/S)_{fppf}$.

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