Write $\mathbb N$ for the natural numbers (with $0$) seen as a monoid under $+$.
Then for any monoid $M$ the underlying set of $M$ can be calculated as $\mathrm{Hom}(\mathbb N,M)$.
But if we view monoids as one object categories, via their deloopings, then $\mathrm{Hom}(\mathrm B\mathbb N,\mathrm BM)$ is a category whose objects are the elements of $M$ and whose morphisms $a\to b$ are elements $c \in M$ with $ca = bc$. Hence the isomorphism classes of this category are the conjugacy classes.
So the conjugacy classes are in this sense a categorified version of the underlying set of $M$.
Now suppose $M$ is commutative. We can now take the double delooping of $M$ to get a $2$-category with one object and only the identity morphism on that object and with $M$ as its $2$-morphisms. Question: What $2$-category is now given by $\mathrm{Hom}(\mathrm B^2\mathbb N,\mathrm B^2M)$?
This should be easy to figure out from the definitions, but every time I try I find that there's simply too much to hold in my head at one time.