Consider the category of finitely presented commutative rings, and equip its opposite category with either the Étale or Zariski topology. These give rise to topoi $\operatorname{Sh}(\mathbf{Et})$ and $\operatorname{Sh}(\mathbf{Zar})$ by taking sheaves. By the descent theorem for schemes, the functors of points of schemes are all sheaves in either of these topoi. In what follows, we will consider the Étale case:

Given a scheme $S$ viewed as an object of $\operatorname{Sh}(\mathbf{Et})$, we can form the slice topos $\operatorname{Sh}(\mathbf{Et})_{/S},$ and by the Yoneda lemma, we see that the functor taking a scheme to this slice topos is fully faithful into the category of categories over $\operatorname{Sh}(\mathbf{Et})$. However, given a map $f:S\to T$ of schemes, the induced functor $$f_!:\operatorname{Sh}(\mathbf{Et})_{/S}\to \operatorname{Sh}(\mathbf{Et})_{/T},$$ is not a geometric morphism. It is however a morphism in $\mathbf{Cat}_{/\operatorname{Sh}(\mathbf{Et})}$ with respect to the projection functors.

Since all of the functors in this triangle admit right adjoints that themselves admit right adjoints, the calculus of mates tells us that we have a triangle of essential (in fact Étale) geometric morphisms:

$$\begin{matrix} \operatorname{Sh}(\mathbf{Et})_{/S} & \xrightarrow{f_*} & \operatorname{Sh}(\mathbf{Et})_{/T}\\ S_* \searrow&\overset{\alpha}{\Leftarrow}&\swarrow T_*\\ &\operatorname{Sh}(\mathbf{Et}) \end{matrix},$$

where the natural transformation $\alpha$ is the mate of the mate of the commuting triangle arising from the Yoneda embedding:

$$\begin{matrix} \operatorname{Sh}(\mathbf{Et})_{/S} & \xrightarrow{f_!} & \operatorname{Sh}(\mathbf{Et})_{/T}\\ S_!\searrow&=&\swarrow T_!\\ &\operatorname{Sh}(\mathbf{Et}) \end{matrix},$$

In particular, since $\operatorname{Sh}(\mathbf{Et})$ is the classifying topos of strictly local rings (also called strictly henselian rings), the geometric morphism $f_*$ together with the 2-cell $\alpha$ determines a map of strictly locally ringed topoi. Since everything is natural and functorial, this defines a functor from schemes (in fact from $\operatorname{Sh}(\mathbf{Et}$)) to the category of strictly locally ringed topoi.

Question: Is this assignment fully faithful in the category of strictly locally ringed topoi?

notfully faithful if $\mathbf{Et}$ is defined, as in your post, using only finitely presented rings. For instance, the functor of points of $\mathrm{Spec}(\mathbb{Q})$ coincides with the functor of points of the empty scheme. $\endgroup$