limits and colimits - Coend computation continued

I agree with Reid's answer, but I want to add a bit more.

Putting Reid's calculation into a more general setting, if $A$ is any category then
$$
int^{a in A} mathrm{hom}_A (a, a)
= (mathrm{endomorphisms in } A)/sim
$$
where $sim$ is the (rather nontrivial) equivalence relation generated by $gh sim hg$ whenever $g$ and $h$ are arrows for which these composites are defined.

You can see confirmation there that your instinct about traces was right. If we wanted to define a 'trace map' on the endomorphisms in $A$, it should presumably satisfy $mathrm{tr}(gh) = mathrm{tr}(hg)$, i.e. it should factor through $int^a mathrm{hom}(a, a)$.

In fact, Simon Willerton has done work on 2-traces in which exactly this coend appears. See for instance these slides, especially the last one.

You can see in that slide something about the dual formula, the end
$$
int_{a in A} mathrm{hom}_A(a, a).
$$
By the "fundamental fact" I mentioned before, this is the set
$$
{}[A, A](mathrm{id}, mathrm{id}) = mathrm{Nat}(mathrm{id}, mathrm{id})
$$
of natural transformations from the identity functor on $A$ to itself. Evidently this is a monoid, and it's known as the centre of $A$. For example, when $G$ is a group construed as a one-object category, it's the centre in the sense of group theory. So your set might, I suppose, be called the co-centre of $A$.