Let S be a scheme. Let (Sch/S) be a small category of schemes over S (including essentially all finitely presented schemes affine over S). Let E = (Sch/S)zar denote the gros Zariski topos with its local ring object A1.
Is there a nice way to construct the petit Zariski topos X = Szar out of the locally ringed topos E? (By nice I mean, for example, that there is a universal property that the locally ringed topos X possesses with respect to E.)
There are variations of this question in which I am also interested: For example, one can replace E by the gros étale (or fppf or fpqc) topos (Sch/S)ét and ask for the construction of Szar out of (Sch/S)ét. Or one can replace X by the petit étale (or fppf or fpqc) topos Sét and ask for the construction of it out of E = (Sch/S)zar.
Best Answer
Many of these toposes admit descriptions as internal classifying toposes, hence indeed enjoy useful universal properties. Here is a selection of such descriptions:
Constructing the big Zariski topos from the little Zariski topos. The big Zariski topos of a scheme $S$ is the externalization of the result of constructing, internally to the little Zariski topos of $S$, the classifying topos of local $\mathcal{O}_S$-algebras which are local over $\mathcal{O}_S$. (One might hope that it would simply be the internal big Zariski topos of $\mathcal{O}_S$, that is the classifying topos of local $\mathcal{O}_S$-algebras where the structure morphism needn't be local. This alternative description is true if $S$ is of dimension $0$.)
Constructing the little Zariski topos from the big Zariski topos. Additionally to $\mathbf{A}^1$, which is the functor $T \mapsto \Gamma(T,\mathcal{O}_T)$, the big Zariski topos contains an additional local ring object: The functor $\flat \mathbf{A}^1$, which maps an $S$-scheme $(f : T \to S)$ to $\Gamma(T,f^{-1}\mathcal{O}_S)$. There is a local ring homomorphism $\flat \mathbf{A}^1 \to \mathbf{A}^1$, and the little Zariski topos can be characterized as the largest subtopos of the big Zariski topos where this morphism is an isomorphism. (This fits with the comments as follows. If $S = \operatorname{Spec}(A)$ is affine, we have the string of ring homomorphisms $\underline{A} \to \flat \mathbf{A}^1 \to \mathbf{A}^1$ starting in the constant sheaf $\underline{A}$. The map $\underline{A} \to \flat \mathbf{A}^1$ is always a localization, and the composition is iff $\flat \mathbf{A}^1 \to \mathbf{A}^1$ is an isomorphism.)
Constructing the big étale topos from the big Zariski topos. The big étale topos of a scheme $S$ is the largest subtopos of the big Zariski topos of $S$ where $\mathbf{A}^1$ is separably closed. This fact is essentially a restatement of Gavin Wraith's theorem on what the big étale topos classifies.
Constructing the big infinitesimal topos from the little Zariski topos. A back-of-the-envelope computation indicates that the recent result of Matthias Hutzler on what the infinitesimal topos of an affine scheme classifies can be relativized to the non-affine case as follows. The big infinitesimal topos of a scheme $S$ is the externalization of constructing, internally to the little Zariski topos of $S$, the classifying topos of local and local-over-$\mathcal{O}_S$ $\mathcal{O}_S$-algebras equipped with a nilpotent ideal.
Some details can be found in Sections 12 and 21 of these notes.
A word of warning: When I say "the largest subtopos where foo", I refer to the largest element in the poset of subtoposes which validate foo from their internal language. (By general abstract nonsense (more or less the existence of classifying toposes), such a largest element always exists in case foo is a set-indexed conjunction of geometric implications.) In particular, I'm not referring to "the subtopos of those objects $Y$ of $E$ such that $\mathbf{A}^1$ restricted to $Y$ enjoys foo" (as in the comments). Indeed, this category is in general not a subtopos (typically it doesn't contain the terminal object). Maybe I interpreted that phrase too literally.