Let $X$ be a scheme. Let $(\mathrm{Sch}/X)_{ét}$ and $X_{ét}$ be the big and small étale sites, resp. Is the restriction functor from $\operatorname{Sh}((\mathrm{Sch}/X)_{ét})$ to $\operatorname{Sh}(X_{ét})$ an equivalence of categories? I'm guessing it isn't, but I haven't been able to come up with a counterexample.
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6$\begingroup$ What examples have you tried? Consider the functor of points of any non-etale $X$-scheme, with $X$ the spectrum of a separable closed field. $\endgroup$– nfdc23Commented Jul 6, 2017 at 1:41
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$\begingroup$ @nfdc23 thanks! that gives an example of the failure of faithfulness, so you should post it as an answer. Do you have examples for the failures of fullness and essential surjectivity? $\endgroup$– Avi SteinerCommented Jul 6, 2017 at 15:56
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1$\begingroup$ Fullness and essential surjectivity are at the level of exercises, so I don't think these should be asked here: by contemplating pullback operations, you can work them out for yourself. Not all questions that come to mind when first learning about a topic should be answered on the Internet. For related reasons, I think it is better to leave my comment as a comment. $\endgroup$– nfdc23Commented Jul 7, 2017 at 0:44
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