Flabby representable sheaves Let $S$ be a scheme. Consider some representable moduli functor $\mathcal{M}:(Sch/S)^{op}\rightarrow Set$ represented by some scheme $M$. Then for each $V\in (Sch/S)^{op}$, let define
$$\mathcal{M}^{glob}(V):=\text{im}\left(\mathcal{M}(S)\rightarrow \mathcal{M}(V)\right).$$
This defines a subfunctor $\mathcal{M}^{glob}\subset \mathcal{M}$. I'm interested if this functor is again representable. I tried showing that this is a open or closed subfunctor, but with no succes.
In a bit more generality, consdier a sheaf representable $\mathcal{F}$ on some site $\mathcal{C}$ with an initial object $X$. Then we can define a maximal flabby sub-pre-sheaf $\mathcal{F}^{glob}$ by defining
$$\mathcal{F}^{glob}(V):=\text{im}\left(\mathcal{F}(X)\rightarrow \mathcal{F}(V)\right).$$
Is this a sheaf again, and is it representable?
 A: Let $\mathcal{C}$ be a small site (or, at least, cofinally small) and let $\textbf{Psh} (\mathcal{C})$ be the category of presheaves on $\mathcal{C}$. There is a functor $\Gamma : \textbf{Psh} (\mathcal{C}) \to \textbf{Set}$ represented by the terminal presheaf (which may or may not be representable in $\mathcal{C}$, at this level of generality), and it has a left adjoint $\Delta : \textbf{Set} \to \textbf{Psh} (\mathcal{C})$ that sends every set $A$ to the "constant" presheaf defined by $(\Delta A) (U) = A$. We have a counit morphism $\epsilon_F : \Delta \Gamma F \to F$ for every presheaf $F$, and your construction is precisely the image of this morphism. Expressed this way, the failure of $\operatorname{Im} \epsilon_F \subseteq F$ to be a sheaf becomes unsurprising: usually we have to sheafify the presheaf image to obtain a sheaf.
If your goal is to construct a flabby (pre)sheaf, then it would be inappropriate to sheafify $\operatorname{Im} \epsilon_F$: sheafification can destroy flabbiness. On the other hand, if we work with presheaves then representability is a rather strong condition: indeed, representable presheaves are projective, so the epimorphism $\Delta \Gamma F \to \operatorname{Im} \epsilon_F$ would be split. But that would make $\operatorname{Im} \epsilon_F$ a retract of a constant presheaf, hence also constant – not very interesting, I think.
Finally, let me remark that the notion of flabby (pre)sheaf does not seem to be appropriate for non-localic sites. The point of flabby sheaves of modules on a topological space or locale is that they are acyclic with respect to the global sections functor, but I don't think this is true for a general site.
