 Subobject-classifier preservation criterion for a sheaf reflector

For a local operator $j$, the following are equivalent: $L$ preserves the subobject classifier; $L(q):L\Omega \to {\Omega }_j$ is an isomorphism; the canonical inclusion $i:{\Omega }_j\hookrightarrow \Omega$ is $j$-dense; and every monomorphism $A^{\prime }\hookrightarrow A$ factors as a $j$-closed mono $A^{\prime }\hookrightarrow A^{\prime \prime }$ followed by a $j$-dense mono $A^{\prime \prime }\hookrightarrow A$.