For nice topological spaces (say Haudorff spaces) $X$ and $Y$, there is a bijection between continuous maps $X\to Y$ and isomorphism classes of geometric morphisms $\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$.


Question: Is there a similar statement for "nice" schemes, i.e., that morphisms of schemes $X\to Y$ correspond bijectively to isomorphism classes of geometric morphisms from the étale topos of $X$ to the étale topos of $Y$?



Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck topoi
Below is a proof that the (pseudo)functor that sends a scheme to its petit étale topos is not fully faithful, for any category of schemes over an algebraically closed base field $k$, assuming that this category contains both $\mathrm{Spec}(k)$ and $\mathrm{Spec}(k[t])$.


*Proof:* The morphisms of schemes
$\mathrm{Spec}(k) \to \mathrm{Spec}(k[t])$ over $k$ correspond to maximal ideals of $k[t]$. At the level of toposes, these give geometric morphisms $\mathbf{Sets} \to \mathcal{E}$, with $\mathcal{E}$ the étale topos of $\mathrm{Spec}(k[t])$. However, not every geometric morphism $\mathbf{Sets} \to \mathcal{E}$ is of this form. There is another one induced by the morphism of schemes $\mathbf{Spec}(k(t)^\mathrm{sep}) \to \mathrm{Spec}(k[t])$, with $k(t)^\mathrm{sep}$ the separable closure of $k(t)$. $\square$


For other reasonable categories of schemes, I expect that the answer to your question will still be "no". However, in practice it might be enough that the functor is "almost fully faithful", see the positive results described in the [answer](https://mathoverflow.net/a/426051/) by D.-C. Cisinki.


