= Solution
Write $h_T(S)=\operatorname{Hom}(S,T)$ for the <functor of points> of a <scheme> $T$. A <subfunctor> $H\subseteq F$ is an <open subfunctor> if every <natural transformation> $h_T\to F$ has a <pullback> represented by an <open subscheme> $U\subseteq T$:
$$
\boxed{h_T\times_F H\cong h_U,\qquad U\hookrightarrow T\text{ an open immersion}.}
$$
The displayed <natural isomorphism> must identify the projection to $h_T$ with the <natural transformation> induced by $U\hookrightarrow T$. Equivalently, for each $\xi\in F(T)$, there is an <open subscheme> $U\subseteq T$ such that a <morphism of schemes> $a:S\to T$ factors through $U$ exactly when $F(a)(\xi)\in H(S)$. The <pullback> is taken in the <functor category>, so this tests arbitrary families, rather than just underlying points.
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