= Solution
The <open affine representability criterion> says that $F$ is a <functor represented by a scheme> exactly when $F$ is a <Zariski sheaf> and admits a family of <open subfunctors> $F_i\subseteq F$, each represented by an <affine scheme>, which covers $F$ in the following sense: for every <scheme> $T$ and every $\xi\in F(T)$, the <open subschemes> representing $h_T\times_FF_i$ cover $T$. In particular, the covering condition does not demand that every $\xi$ lie in one $F_i(T)$.
In symbols, for <affine schemes> $X_i$,
$$
\boxed{F_i\cong h_{X_i},\quad F_i\subseteq F\text{ open},\quad \bigcup_i(T\times_FF_i)=T,\quad F\text{ a Zariski sheaf}.}
$$
The family is indexed by a set. For necessity, take an <affine open cover> of a representing <scheme>; <morphisms of schemes> glue over a <Zariski cover>. For sufficiency, $F_i\times_FF_j$ is represented by an <open subscheme> of each $X_i,X_j$. The two descriptions identify these overlaps and satisfy the compatibility on triple overlaps because all maps come from the inclusions into $F$. Apply <gluing of schemes along open subschemes> to obtain $X$. Local identification gives $h_X\to F$, and the <Zariski sheaf> condition makes this a <natural isomorphism>: its elements can be constructed and compared on the pulled-back <open cover>.
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