I am having trouble proving a result in Mumfords book 'Lectures on Curves on an Algebraic surface.

It is a statement about the representability of some functor. It is stated on page 108 and says the following. Let $k$ be a field (does anything change if we consider a noetherian scheme instead?) Let $G$ be a scheme over $k$ and $B$ a functor from the category of schemes over $k$ to sets. Let $A$ be a sub functor of $B$. Consider the following commutative diagram of functors

$$ \require{AMScd} \begin{CD} A@>>\Phi> h_G\\ @VVV @| \\ B @<<\Psi<h_G \end{CD} $$ where the first vertical map is the natural inclusion as sub functors and the right vertical map is the identity.

Now assume that for each scheme $S$ over $k$. And for all $\alpha \in B(S)$, there is a subschema $Y\subset S$ such that all $g\colon T\to S$ the pullback $g^*(\alpha)\in B(T)$ lies in the subset $A(T)$ if and only if $g$ factors through $Y$.

Then he states that $A$ is representable by some subschema $G_0\subset G$.

Well I guess the prove works by setting $S=G$ and $G_0=Y$. Then $\Phi$ factors through $h_{G_0}$. But I do not see why this is a surjective mapping?

Does the proof work differently or is this general statement just wrong?