= Solution
A <local operator>, also called a <Lawvere-Tierney topology>, is a map $j:\Omega\to\Omega$ which internally satisfies
$$
p\leq jp,\quad j\top=\top,\quad j(p\wedge q)=jp\wedge jq,\quad jjp=jp.
$$
The <closure operation of a local operator> sends a mono with characteristic map $\chi$ to the <subobject> classified by $j\chi$. It is inflationary, idempotent and <pullback>-stable. A mono is <j-dense> if its closure is its whole codomain, and <j-closed> if it equals its closure. A <j-sheaf> $S$ is an object for which restriction
$$
\mathcal E(B,S)\longrightarrow\mathcal E(A,S)
$$
is a bijection for every j-dense mono $A\hookrightarrow B$; requiring only injectivity defines a <j-separated object>.
Here is a construction underlying the <sheaf reflector for a local operator>. The <closed-subobject classifier> $\Omega_j=\{p:jp=p\}$ is a j-sheaf: closed <subobjects> on a dense <subobject> extend uniquely by taking their closure in the larger object. Powers $\Omega_j^X$ are also sheaves, because products of a dense mono with $X$ remain dense. A j-closed <subobject> of a sheaf is a sheaf: first extend a map into the ambient sheaf, then use density to force its image into the closed <subobject>.
Close the diagonal of $X$. Its j-closure is an equivalence relation, using preservation of finite meets and <pullback>-stability to verify transitivity. The effective quotient $X\twoheadrightarrow X_s$ is the separated reflection: every map from $X$ into a separated object identifies that closed diagonal and factors uniquely. For separated $X_s$, the closed-singleton map
$$
X_s\longrightarrow\Omega_j^{X_s},\qquad x\longmapsto\bigl(z\longmapsto j(z=x)\bigr)
$$
is monic. Its j-closed image closure $a_jX$ is a sheaf, and $X_s\hookrightarrow a_jX$ is dense. Unique extension across that mono, following the separated quotient factorization, proves
$$
\mathcal E(X,S)\cong\mathbf{sh}_j(\mathcal E)(a_jX,S)
$$
for every sheaf $S$. This proves reflectivity. The closure construction is <pullback>-stable; equivalently, separated quotients and the subsequent dense embeddings commute with the finite limiting comparisons, giving the usual left-exact sheaf reflector.
<Finite limits> of sheaves are computed in $\mathcal E$, because unique extensions can be taken componentwise. If $S$ is a sheaf, $S^T$ is a sheaf for any $T$, by the same product-with-dense-mono argument. Thus sheaf exponentials are the ambient exponentials. Monos between sheaves have j-closed images: their closure is a sheaf, and the dense inclusion into it splits by the extension property, hence is an isomorphism. Therefore $\Omega_j$ classifies precisely their <subobjects>. These observations establish \b[$\mathbf{sh}_j(\mathcal E)$ is a reflective topos].
Now let $u:1\to\Omega$ classify the given <subterminal object>. Its <open local operator> and <closed local operator> are
$$
\boxed{o(U)(p)=(u\Rightarrow p),\qquad c(U)(p)=u\vee p.}
$$
The <Heyting algebra> identities verify all local-operator axioms: implication by fixed $u$ preserves meets and is idempotent, while adjoining $u$ preserves meets by distributivity and is idempotent.
For a mono in $B$ with characteristic predicate $p$, closedness for $c(U)$ means $u\vee p=p$, or $u\leq p$. Density for $o(U)$ means $(u\Rightarrow p)=\top$, again $u\leq p$. Thus \b[the c(U)-closed monos are exactly the o(U)-dense monos].
Both densities together force $u\leq p$ and $u\vee p=\top$, hence $p=\top$: the only jointly dense monos are isomorphisms. More explicitly, the meet of these operators is pointwise and
$$
(u\Rightarrow p)\wedge(u\vee p)=p,
$$
so $o(U)\wedge c(U)=\mathrm{id}_\Omega$.
For their join, every mono $A\hookrightarrow B$ factors through the union with the <pullback> $U_B=U\times B$:
$$
A\hookrightarrow A\cup U_B\hookrightarrow B.
$$
The first mono is c(U)-dense, because adjoining $U$ fills its codomain; the second is o(U)-dense, because its image contains $U_B$. Any <local operator> above both must therefore make every mono dense, since its dense monos are closed under composition. It is the largest operator $p\mapsto\top$. Consequently
$$
\boxed{o(U)\wedge c(U)=\mathrm{id}_\Omega,\qquad o(U)\vee c(U)=\top.}
$$
These are the <complementary open and closed local operators> in the ordered lattice of <local operators>, with order given by pointwise implication.
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