I feel like this should be fairly simple, but I can't quite see how to answer this question: in a category with products and coequalizers, what is the coequalizer of the pair $\pi_1,\pi_2:A\times A\rightrightarrows A$?
My gut feeling says it's the terminal object (it's the case in $\textbf{Set}$), but I'm not sure it even exists in general. Forgive me if this is a basic category theory exercise I never saw.
Welcome to MSE ^_^
If a terminal object $1$ exists, and moreover there is a map $a : 1 \to A$, then yes. The coequalizer is $1$.
Notice we can't drop the existice of a map $1 \to A$ even in $\textbf{Set}$! The empty set modulo the empty relation is not $1$.
But, if $a : 1 \to A$ exists, then it's not hard to show that for any $f : A \to X$ making $f \pi_1 = f \pi_2$ we must also have $f a_1 = f a_2$ for any $a_1, a_2 : 1 \to A$.
Once we take this into account, it is clear that the map $f a : 1 \to X$ makes the coequalizer diagram commute for any choice of arrow $a : 1 \to A$, and we assumed one existed.
I hope this helps ^_^