3.18 in Johnstone's Topos Theory is the proof that the classifying map of a subobject $J\hookrightarrow\Omega$ is a Lawvere-Tierney topology if and only if the class $\Xi_J$, of monomorphisms whose classifying maps factor through $J$, is closed under composition and contains all isomorphisms. Near the beginning of the proof of the "if" direction he makes the following remark:
Then it is easily seen that the operation $\rho$ on subobjects induced by $j$ [the classifying map of $J$] is describable as follows: if $X'\rightarrowtail X$ is a subobject of $X$, then $\rho(X')$ is the unique largest $X''\rightarrowtail X$ for which $X'\cap X''\rightarrowtail X''$ is in $\Xi_J$.
How do I prove that this is the case?
My reading here is that $\rho(X')$ is the subobject classified by $j\circ\chi_{X'}$ (where $\chi_{X'}:X\to\Omega$ classifies $X'$). I can't really show my work so far because it's several pages worth of confusing diagrams, and I'm entirely turned around at this point. I feel like the right approach must involve the fact that $\Xi_J$ is stable under pullbacks and pushouts, but I haven't seen how to use this yet. I have recently convinced myself again why $X'\cap\rho(X')\rightarrowtail\rho(X')$ is in $\Xi_J$, but the maximality of $\rho(X')$ w.r.t. this property still eludes me.
This is much simpler than I was making it.
If $\chi_{X'}:X\to\Omega$ classifies $X'\rightarrowtail X$, and $i:X''\rightarrowtail X$ is any subobject, then $\chi_{X'}\circ i$ classifies $X'\cap X''\rightarrowtail X''$. So for the classifying arrow of $X'\cap X''\rightarrowtail X''$ to factor through $J$ means that there is a commutative diagram $$\begin{array}{ccc} X'' & \rightarrow & J \\ i \downarrow & & \downarrow\\ X & \underset{\chi_{X'}}{\rightarrow} & \Omega \end{array}$$ In the case of $\rho(X')$, its definition as the subobject classified by $j\circ\chi_{X'}$ means that the top arrow in the diagram $$\begin{array}{ccc} \rho(X') & \rightarrow & J \\ \downarrow & & \downarrow\\ X & \underset{\chi_{X'}}{\rightarrow} & \Omega \end{array}$$ exists to make the square commute, and that the square is actually a pullback; from which it is immediate that $X'\cap\rho(X')\rightarrowtail\rho(X')$ is in $\Xi_J$ and that $\rho(X')$ is the maximal such subobject of $X$ for which this holds.