Suppose that $\mathcal{C}$ is a category of sets and functions and $F:\mathcal{C}\to Set$ is a presheaf.
We can define a subfunctor of $F$, $F^*$, as follows:
$F^*(A) := \{a \in F(A)\mid$ for some finite $Y\subseteq A$ for any $B$ in $\mathcal{C}$ and $h,h':A\to B$, $F(h)a=F(h')a$ whenever $hy=h'y$ for all $y\in Y\}$.
$F^*(h):=F(h)$
Intuitively this corresponds to the finitely definable elements of $F$, in the sense that a finite bit of data from $A$ fixes the behavior of the elements of $F^*(A)$. I've added an example in the comment below.
Does this sort of construction have a name (or is it subsumed by a more general notion)? Looking for further connections and references here.