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.