What's the last step in this proof of the uniqueness of equalizers?

I’m having a mind-blank right now, and I can’t seem to find an explanation of this in the texts I’ve consulted. It’s to do with the uniqueness up to isomorphism of equalizers.

The proof goes:

Let $e: E \rightarrow A$ and $e^\prime : E^\prime \rightarrow A$ be equalizers of morphisms $f$ and $g$ from $A$ to $B$. Then there is a unique $k : E^\prime \rightarrow E$ and a unique $h:E\rightarrow E^\prime$ such that
$$e^\prime = e \circ k$$
and
$$ e = e^\prime \circ h,$$
thus
$$e^\prime = e \circ k = e^\prime \circ h\circ k$$
and
$$ e = e^\prime\circ h = e \circ k \circ h. $$

My question is:

What then do we invoke to say that this implies $h\circ k$ and $k\circ h$ are the identity morphisms?

Adamek-Herrlick-Strecker in their Abstract and Concrete Categories (from which this is taken) merely say “by the uniqueness requirement” in the definition of equalizers, which I do not see how to apply.

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