# Why is an epic equalizer an isomorphism?

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 $k \circ e = id_X$?

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$id_A$ is an equalizer of $f$ and $g$.