= Solution
Every characteristic map $a:X\to\Omega$ expresses its <subobject> as $a^*\top$. Universality of the <closure operation induced by a left-exact reflector> gives
$$
c_X(a^*\top)=a^*c_\Omega(\top).
$$
By definition, $j$ classifies $c_\Omega(\top)$, so the right-hand subobject has characteristic map $j\circ a$. The original and its closure agree exactly when their unique characteristic maps agree. Thus
$$
\boxed{a\text{ classifies a closed subobject}\Longleftrightarrow j\circ a=a.}
$$
Extensivity, idempotence, preservation of intersections and closure of the whole subobject also give the <Lawvere-Tierney topology> laws for $j$.
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