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.