Let $\mathcal{C}$ be a regular category, $A\in\mathcal{C}$ and suppose the two canonical projections $A\times A \rightrightarrows A$ are equal. I'd like to show that then $A\cong 1$, where $1$ is the terminal object of $\mathcal{C}$. This is a step in the proof of Lemma 1.3.8 in The Elephant, and I can't see why its true. Showing that $A$ must be terminal seems difficult, since there's no obvious way of getting an arrow $X\to A$ for every $X\in\mathcal{C}$.
For reference, here's the actual result:
