(co)reflector to the forgetful functor $U:\mathbf{CMon} \to \mathbf{ Mon}$ I've been asking questions on reflectors before and I hope you are not getting annoyed. Apologies if that's the case.
My question is the following:
Are there reflectors to the forgetful functor $U: \mathbf{CMon} \to \mathbf{Mon}$ from commutative monoids to the general monoids? 
I know they exist in rings and groups but I have trouble working it out for monoids. Any answer is very much appreciated, but one not referring to the adjoint functor theorem is preferred.

Well it's pretty much the same as for groups, and in fact, every algebraic structure which includes a binary composition law.
When $M$ is a monoid, then $M^{\mathrm{ab}}$ is defined to be $M/\sim$, where $\sim$ is the smallest congruence relation on $M$ which satisfies $ab \sim ba$ for all $a,b \in M$. Then $M \mapsto M^{\mathrm{ab}}$ is left adjoint to the inclusion functor $U$. The proof is trivial.
If you want to have an explicit description of $\sim$ (which is important for computations, but not for the proof that the above is true): $x \sim y$ iff there is a composition $x=x_1 \cdots x_n$ and a permutation $\sigma$ of $1,\dotsc,n$ such that $y = x_{\sigma(1)} \cdots x_{\sigma(n)}$. [This easy description is not available for the category of groups] In other words: Computation in $M^{\mathrm{ab}}$ is as in $M$, but you don't care for the order in which they are done.
