 Solution

For a subobject $m:S\hookrightarrow X$, define the closure operation induced by a left-exact reflector bywhere the pullback uses ${\eta }_X:X\to LX$ and the monomorphism $Lm:LS\hookrightarrow LX$. Naturality of the unit factors $S\hookrightarrow X$ through $c_X(S)$, proving extensivity, and functoriality of $L$ proves monotonicity. Applying the left-exact $L$ to the defining pullback gives $L(c_X(S))\cong LS$, since $L{\eta }_X$ is invertible. Repeating the construction therefore changes nothing, proving idempotence.

For $h:Y\to X$, naturality of $\eta$ and preservation of pullbacks identify the closure of $h^{\ast }S$ with $Y{\times }_{LX}LS$, which is also $h^{\ast }{c}_X(S)$. HenceThe operation is thus universal: it is stable under every pullback. The same construction preserves finite intersections because $L$ is left exact.