Pullback of a Zariski Sheaf to the Etale SIte.

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Suppose $F$ is a sheaf on the big etale site of a scheme $S$ (i..e, where the objects are all schemes over $S$). If we restrict $F$ to the small Zariski site of $S$ and then pull it back to the small etale site, do we get the restriction of $F$ to the small etale site?