 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.

ConsequentlyReplacing $X$ 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 becomesWith the factored truth arrow this is precisely the subobject classifier universal property. Thus the closed-subobject classifier supplies the classifier in the reflective subcategory.