 Solution

Every characteristic map $a:X\to \Omega$ expresses its subobject as $a^{\ast }\top$. Universality of the closure operation induced by a left-exact reflector givesBy 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. ThusExtensivity, idempotence, preservation of intersections and closure of the whole subobject also give the Lawvere-Tierney topology laws for $j$.