Closure operation induced by a left-exact reflector ($c_A$)

For a subobject $m:A^{\prime }\hookrightarrow A$ and a left-exact reflector $L$ with unit $\eta$, define $c_A(A^{\prime })$ by the pullback in a categoryThis operation is monotone, inflationary, idempotent, and stable under pullback. If $A$ is fixed by $L$, then $A^{\prime }$ is fixed by $L$ exactly when $A^{\prime }$ is closed.