Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck topoi 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 $Y$?
 A: Take a look at the paper of Barwick, Glasman and Haine (https://arxiv.org/pdf/1807.03281.pdf).
In particular the section "Exodromy for schemes & the Reconstruction Theorem".
A: There is no way it is true in general, but there are results in this direction nevertheless: Theorem 3.1 in this paper of Voevodsky establishes (a kind of fully) faithfulness for normal schemes of finite type over a number field, in the spirit of a conjecture of Grothendieck (in his letter to Falting, an English translation of which is published in Geometric Galois actions, 1, ser. London Math. Soc. Lecture Note Ser. Vol. 242, 1997, and in his Esquisse d'un programme).
In particular, if $X$ and $Y$ are normal schemes of finite type over a number field $K$ (in fact any field of finite type of characteristic zero) and if their small étale topoi are equivalent over the small étale topos of $K$, then $X\cong Y$ as $K$-schemes.
There are also Torelli-like theorems saying that one can sometimes determine the isomorphism class of a scheme from its topological space only (i.e. from the 0-localic topos associated to its étale topos). This paper of János Kollár, Max Lieblich, Martin Olsson, Will Sawin shows that, for $K$ an uncountable algebraically closed field of characteristic zero $K$, one can reconstruct a normal geometrically integral proper scheme of dimension at least 2 from its underlying topological space. Their results also prove, for $K$ an uncountable algebraically closedfield of characteristic zero, one can reconstruct a proper normal $K$-scheme of dimension at least 2 from its category of constructible abelian étale sheaves.
A: 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 by D.-C. Cisinki.
