= Closure operation induced by a left-exact reflector
{title2=$c_A$}
For a subobject $m:A'\hookrightarrow A$ and a left-exact reflector $L$ with unit $\eta$, define $c_A(A')$ by the <pullback in a category>
$$
\begin{array}{ccc}
c_A(A')&\longrightarrow&LA'\\
\downarrow&&\downarrow Lm\\
A&\xrightarrow{\eta_A}&LA.
\end{array}
$$
This operation is monotone, inflationary, idempotent, and stable under pullback. If $A$ is fixed by $L$, then $A'$ is fixed by $L$ exactly when $A'$ is closed.
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