ag.algebraic geometry - are closed subfunctors complimentary to open subfunctors?

I apologize if the following question has already been asked and settled. I couldn't find any thread.

Say, $mathcal{C} = (Sch/k)$, the category of schemes over $k$ (a field). Let $mathcal{F} in mathcal{C}^{wedge}$, be an object of $mathcal{C}^{wedge}$ - the category of contravariant functors from $mathcal{C}$ to $(Sets)$. One has the set of points:

$$ |mathcal{F}| := lim_{to} mathcal{F} (K), $$

the limit taken over fields $K/k$. Given a subfunctor $mathcal{G} hookrightarrow mathcal{F}$ one gets a subset $|mathcal{G}| subset |mathcal{F}|$ (ie. a canonical map from $|mathcal{G}| to |mathcal{F}|$ that is injective). In particular, $|mathcal{U}|$ for the open subfunctors $mathcal{U} hookrightarrow mathcal{F}$ form a topology on $|mathcal{F}|$.

Question: Given a closed subset $Z subset |mathcal{F}|$ does there exist a closed subfunctor (possibly non-unique)
$mathcal{Z} hookrightarrow mathcal{F}$ so that $Z = |mathcal{Z}|$ (as subsets of $|mathcal{F}|$)?

In some sense, are open subfunctors and closed subfunctors really "complimentary"?