= Solution
Let $\Omega_j\hookrightarrow\Omega$ be the <equalizer> of $j$ and the identity. By part (iv), maps $X\to\Omega_j$ correspond exactly to closed <subobjects> of $X$. Closed subobjects of $X$ in turn correspond to <subobjects> of $LX$ in $\mathcal L$: send $S$ to $LS\hookrightarrow LX$, and in the reverse direction pull a fixed subobject $N\hookrightarrow LX$ back along $\eta_X$. Left exactness shows that these operations are inverse; the pulled-back object has reflection $N$, making it closed by the defining formula.
Consequently
$$
\mathcal E(X,\Omega_j)\cong\operatorname{Sub}_{\mathcal L}(LX).
$$
Replacing $X$ by $LX$ and using $L\eta_X$ invertible shows that maps into $\Omega_j$ extend uniquely across every unit. The criterion of part (i) therefore proves $\Omega_j$ itself is fixed, so belongs to $\mathcal L$. Truth factors through it because $j\top=\top$. For fixed $X$, the displayed bijection becomes
$$
\boxed{\mathcal L(X,\Omega_j)\cong\operatorname{Sub}_{\mathcal L}(X)}.
$$
With the factored truth arrow this is precisely the <subobject classifier> universal property. Thus the <closed-subobject classifier> supplies the classifier in the reflective subcategory.
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