Category of copresheaves over commutative monoids

Question

Let C be a symmetric monoidal category. Let Comm(C) be the category of commutative monoids in C. Consider the topos X = CoPSh(Comm(C)) of covariant functors from Comm(C) to the category Set of sets.

Which extra data do we have to specify on the topos X such that we can recover (up to some notion of equivalence) the essential structure of the underlying site? For example, every representable copresheaf F = Hom(A, _) has a category of modules attached to it, which is a kind of extra data. [I know I am a bit unprecise here with what I mean by essential, so making this precise could also be part of the answer to my question.]

One could also rephrase this question as follows: Given a topos X that satisfies Giraud's axioms, one can extract a site such that X is the Grothendieck topos over this site. Which extra data do we need to impose on X such that we can recover X as a Grothendieck topos over a site that is (the dual) to commutative monoids in a symmetric monoidal category.

When I write down this question, I have the following example in mind: Let C be the category of abelian groups. Then X is the topos of presheaves on the category of affine schemes, which gives rise to algebraic geometry. X possesses a commutative ring object, namely the affine line A¹ and one has the stack of categories of quasi-coherent sheaves over objects of X. Taking the idea of (Grothendieck) topoi seriously, one should be able to forget about C and just consider the topos X (i.e. without a fixed base site). Of course, one has to remember (at least) A¹. This allows to recover the stack of categories of quasi-coherent modules.

Added for clarification:

But what if C is not the category of abelian groups? In this case, X = CoPSh(Comm(C)) also carries a stack QCoh of categories of quasi-coherent modules as follows: Let F be an object of X, i.e. F is a covariant functor from Comm(C) to Set. An object M of the category QCoh(F) maps a morphism a: Hom(A, _) -> F to an A-module M(a) in C together with natural isomorphisms M(b) = M(a) ⊗_A B for all morphisms a -> b in Comm(C). [It is here where the category C itself comes in.]

What is the minimal amount of data we need on X so that X is equivalent to sheaves on a site S such that the dual of S is of the form Comm(C') with C' giving rise to a somewhat equivalent stack of categories of quasi-coherent modules.

Answer 1

I'm not sure if this is part of your question, but I just want to observe that there's no hope of recovering C from Comm(C), let alone from X = CoPSh(Comm(C)).

For example, take any commutative monoid and regard it as a discrete symmetric monoidal category C. In other words, the objects of the category are the elements of the monoid, the only morphisms are identities, and the tensor product in the category is the multiplication in the monoid. Then there is precisely one commutative monoid in C (namely, the unit object), and it has no endomorphisms. Hence Comm(C) is the terminal category. This is true for whatever commutative monoid you started with. So C can't be recovered from Comm(C).

Or were you only trying to recover Comm(C), not C, from X?