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
$$e^\prime = e \circ k$$
$$ e = e^\prime \circ h,$$
$$e^\prime = e \circ k = e^\prime \circ h\circ k$$
$$ e = e^\prime\circ h = e \circ k \circ h. $$
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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.
Consider the diagram formed from the morphisms $e : E \rightarrow A$ and another copy of $e: E \rightarrow A$. By definition, there is a unique morphism $E \rightarrow E$ which makes this diagram commute. But you proved that $k \circ h$ and the identity are two such morphisms, so they are equal. Same for $E’$.