Forgive the basic question (and the typesetting!) I'm a relative novice regarding category theory, but I've recently decided to teach myself at least the rudiments of toposes. Having stumbled upon Tom Leinster's excellent introductory paper, I'm getting most of the concepts, but I'm still a little woolly on category theoretic proof, and I've been struggling with the following exercise (on page 4 of the paper)
Let $\mathcal{E}$ be a category and let $t: T \to \Omega$ be a mono in $\mathcal{E}$. Suppose that for every mono $ m: A \to X$, $X$ in $\mathcal{E}$ , there is a unique map $\chi :X \to \Omega$ such that there is a pullback square
Then $T$ is terminal in $\mathcal{E}$ .
It's pretty clear that the uniqueness of the map $f: A \to T$ is going to be a consequence of the uniqueness of the map guaranteed by the universal property of pullbacks- perhaps taking the another map $m': A \to X$, and a map $g:A \to T$, and showing from the universal property that $f=g$, but I can't seem to shake the dependency on $m'$. Are there other tricks to be played? Perhaps, does one set $X=A$ and $m=\operatorname{id}_A$?