I've seen in several places without further comment that if an equalizer is epic, it's an isomorphism. I've only proved one half of this:
Suppose $e:X \rightarrow A$ is an epimorphism and an equalizer for $f$ and $g$. Then $f \circ e = g \circ e \implies f = g$. Then any function $e': X' \rightarrow A$ trivially equalizes $f$ and $g$, so take $id_A: A \rightarrow A$. $e$ is an equalizer, so there exists a unique $k: A \rightarrow X$ such that $e \circ k = id_A$.
That gets me one side of the inverse, but how do I prove that $k \circ e = id_X$?