# Comparison between Etale and Zariski topology on schemes

Let $Sch_{Zar}, Sch_{et}$ denote scheme with Zariski and Etale topology respectively. Is there a functor from $Sch_{et}$ to $Sch_{Zar}$ (or from $Sch_{Zar}$ to $Sch_{et}$) which preserves fiber product? Furthermore, can we define a morphism of schemes from $X_{et}$ to $X_{Zar}$ where $X$ is a scheme? If so what are the properties of this morphism?

Any suggestions on reference where similar questions are studied will be most helpful.

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I don't think this is a very well posed question. For one, you probably don't mean a morphism of schemes from $X_\text{ét}$ to $X_\mathrm{Zar}$ but a morphism of sites.