Exercise on "locality" in topos theory

Question

Let $\mathcal E=\mathsf{Sh}(\mathsf C,J)$. Let $A\rightarrowtail \Omega$ be a fixed subobject. For each $X$ in $\mathcal E$, define $T_A(X)$ to be a set of subobjects of $X$ as follows. $U\rightarrowtail X$ is in $T_A(X)$ if its characteristic arrow factors through $A\rightarrowtail X$.

I'm trying to prove the following. Suppose $(U_i\to X)$ is a $J$-covering family. If the pullback of $U\rightarrowtail X$ along each of the $U_i\to X$ is in $T_A(U_i)$, then $U\rightarrowtail X$ is in $T_A(X)$.

However, I'm not really getting anywhere. Even if I assume the site is superextensive, I'm still stuck with rectangular diagrams whose exterior and left square are pullbacks, and for such diagrams, the right square is a pullback iff the bottom left arrow is a universal strong epimorphism.

How can I prove this claim?

Answer 1

Let $ \chi : X \rightarrow \Omega$ be the characteristic function of $U$.

By definition of a subobject classifier, the characteristic function of the pullback of $U$ by $U_i \rightarrow X$ is just the composite $U_i \rightarrow X \overset{\chi}{\rightarrow} \Omega$.

Hence the proposition you are trying to prove boils down to the following fact:

if for all $i$, the map $ U_i \rightarrow X \overset{\chi}{\rightarrow} \Omega$ factor into $A \subset \Omega$ then $\chi$ factor into $A$, and this just the fact that $A$ is a sheaf.