Lemma 59.103.1. 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 $(\mathit{Sch}/S)_ h$.
Proof. The statement about the étale topology is the content of Lemma 59.39.2. To finish the proof it suffices to show that $\pi _ S^{-1}\mathcal{F}$ is a sheaf for the h topology. However, in Lemma 59.102.1 we have shown that $\pi _ S^{-1}\mathcal{F}$ is a sheaf even in the stronger ph topology. $\square$
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