= Solution
Let $\mathcal E$ be an <elementary topos> with <subobject classifier> $\top:1\to\Omega$. A <Lawvere-Tierney topology>[local operator] is a map $j:\Omega\to\Omega$ which is inflationary, idempotent, preserves truth, and preserves binary meets:
$$
p\leq j(p),
\qquad j(jp)=j(p),
\qquad j(\top)=\top,
\qquad j(p\wedge q)=j(p)\wedge j(q).
$$
If $m:A'\hookrightarrow A$ is classified by $\chi_m:A\to\Omega$, its <closure operation of a local operator> is classified by $j\chi_m$. The mono is <j-dense monomorphism> when this closure is all of $A$, and $j$-closed when it equals its closure. An object $X$ is a <j-sheaf> when every map $A'\to X$ along a $j$-dense mono $A'\hookrightarrow A$ extends uniquely to $A\to X$.
The <closed-subobject classifier> is the equalizer
$$
i:\Omega_j\hookrightarrow\Omega
\mathrel{\substack{\xrightarrow{\mathrm{id}}\\[-2pt]\xrightarrow[j]{} }}\Omega.
$$
Thus maps to $\Omega_j$ classify precisely the $j$-closed subobjects. Idempotence factors $j$ as
$$
\Omega\xrightarrow{q}\Omega_j\xrightarrow{i}\Omega,
\qquad qi=1_{\Omega_j},
\qquad iq=j.
$$
To prove that $\Omega_j$ is a $j$-sheaf, let $m:B\hookrightarrow A$ be dense and let $f:B\to\Omega_j$ classify a closed subobject $C\hookrightarrow B$. Take the closure in $A$ of the composite $C\hookrightarrow A$. Pullback stability of closure gives
$$
\overline C^{,A}\cap B=\overline C^{,B}=C,
$$
so the classifier $A\to\Omega_j$ of $\overline C^{,A}$ extends $f$. If two closed subobjects of $A$ restrict to the same subobject of dense $B$, the equalizer of their classifiers is a closed subobject containing $B$; it is both closed and dense and hence equals $A$. The extension is therefore unique.
Let
$$
L:\mathcal E\longrightarrow\mathbf{sh}_j(\mathcal E)
$$
be the <sheaf reflector for a local operator>. The subobject classifier in the sheaf topos is $\Omega_j$. We prove the four assertions through the cycle
$$
(i)\Longleftrightarrow(ii)\Longleftrightarrow(iii)\Longleftrightarrow(iv).
$$
The canonical map comparing the reflected ambient classifier with the sheaf classifier is
$$
L(q):L(\Omega)\longrightarrow L(\Omega_j)\cong\Omega_j.
$$
Consequently $L$ preserves the subobject classifier exactly when $L(q)$ is an isomorphism. This proves $(i)\Longleftrightarrow(ii)$.
Since $qi=1_{\Omega_j}$, one has
$$
L(q)L(i)=1_{\Omega_j}.
$$
If $L(q)$ is an isomorphism then $L(i)$ is its inverse. Conversely, if $L(i)$ is an isomorphism, the same equation makes $L(q)$ its inverse. A monomorphism is sent to an isomorphism by sheafification exactly when it is $j$-dense, so $(ii)\Longleftrightarrow(iii)$.
Assume $(iii)$ and let $m:A'\hookrightarrow A$ have characteristic map $\chi:A\to\Omega$. Form the <pullback in a category>[pullback]
$$
\begin{array}{ccc}
A''&\longrightarrow&\Omega_j\\
\downarrow d&&\downarrow i\\
A&\xrightarrow{\chi}&\Omega.
\end{array}
$$
Because dense monos are pullback-stable, $d:A''\hookrightarrow A$ is $j$-dense. The original $A'$ factors through $A''$, and its characteristic map inside $A''$ is the top horizontal map followed by $i$, which is fixed by $j$. Hence $A'\hookrightarrow A''$ is $j$-closed. This proves $(iv)$.
Finally assume $(iv)$ and apply it to $i:\Omega_j\hookrightarrow\Omega$:
$$
\Omega_j\xrightarrow{c}A''\xrightarrow{d}\Omega,
$$
where $c$ is closed and $d$ is dense. Since $d$ is monic and $i=dc$, the map $c$ is the pullback of $d$ along $i$. It is therefore dense as well as closed, and hence is an isomorphism. Thus $i$ is, up to an isomorphism, the dense mono $d$, proving $(iii)$. All four conditions are equivalent, as summarized by the <subobject-classifier preservation criterion for a sheaf reflector>.
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