I am beginner in the theory of Grothendieck topologies and I have the following question.
Let $X, Y$ be schemes over an algebraically closed field $k$. Denote by $X\_{et}$ and $Y\_{et}$ the Etale sites on $X$ and $Y$, respectively. Suppose that there is a continous morphism from $X\_{et}$ to $Y\_{et}$. Is there any additional conditions under which we can get a morphism (as schemes) from $X$ to $Y$?



Morphism on schemes induced by continuous morphism of sites
There is a full and faithful embedding of the category of schemes into the $2$-category of (*edit: stricly Henselian ringed*) topoi, which sends each scheme $X$ to the topos $Sh\left(X\_{et}\right)$ (in fact, this is fully faithful even on Delign-Mumford stacks, see: <https://eudml.org/doc/90454> Theorem 50, and Definition 43). So, given a geometric morphism $Sh\left(X\_{et}\right) \to Sh\left(Y\_{et}\right)$ respecting structures sheaves is the same as given a morphism of schemes $X \to Y$. In particular, any functor $X\_{et} \to Y\_{et}$ which induces a geometric morphism (*edit: respecting structure sheaves*), also induces a morphism of schemes, but not every geometric morphism can be realized as a morphism of sites.


Now, the structure sheaves are realized as a geometric morphism $Sh\left(X\_{et}\right) \to \mathscr{T},$ with $\mathscr{T}$ the topos of sheaves on the site $Ftyp$ of all schemes of finite type, with the etale topology. This is the classifying topos for strictly Henselian rings. In fact, we can enlarge $Ftyp$ to all schemes locally of finite type $LFtyp$ without changing the topos. If $X$ is also locally of finite type, there is a canonical functor $\pi\_X:X\_{et} \to LFtyp$ sending each etale map $X' \to X$ to $X'.$ This is a morphism of sites which induces the geometric morphism $Sh(X\_{et}) \to \mathscr{T}$ encoding the structures sheaf. In particular, any map of sites $f:X\_{et} \to Y\_{et}$ commuting over their projections to $LFtyp$ encodes a morphism $X \to Y.$ However, this is too strict, since maps of ringed topoi need not respect structure sheaves up to isomorphism, but only induced a map (like in ringed spaces), i.e. it suffices to have a natural transformation $\alpha:\pi\_Y \circ f \to \pi\_X$.


