An exercise in Goldblatt's "Topoi" goes as follows, assuming the existence of set-indexed copowers of 1:
Let $\{ a_s \rightarrowtail 1 : s \in S \}$ be a set of subobjects of $1$ in $\mathcal{E}$, with characteristic arrow $\chi_s : 1 \rightarrow \Omega$ for each $s \in S$. Show that the support of the subobject whose characteristic arrow is the coproduct of the $\chi_s$'s is a join of the $a_s$'s in $\text{Sub}(1)$.
I fail to prove it even in the simple, finite case. So suppose we have $\alpha : a \rightarrowtail 1$ and $\beta : b \rightarrowtail 1$, and let $z \rightarrowtail 1 + 1$ be the pullback of $[\chi_\alpha, \chi_\beta] : 1 + 1 \rightarrow \Omega$. Now we want to prove that $\text{sup}(z) \cong a \cup b$.
Hopefully I understood what's needed right! Now, the main direction of attack I've chosen is to either show the existence of an epic $a + b \twoheadrightarrow \text{sup}(z)$ or $z \twoheadrightarrow a \cup b$ — then the uniqueness of epi-monic factorization would imply the required isomorphism.
But, no matter what I did I couldn't prove the existence of any such arrows. In fact, drawing a bunch of pullbacks and coproducts I constructed an arrow $a + b \rightarrow z$ (as the coproduct arrow of arrows $a, b \rightarrow z$, in turn arising from $z$ being a pullback and certain diagrams commuting), but I also failed to prove it's epic.
So, how do I prove at least the finite case?