In trying to begin to understand the idea of a $k$-tuply monoidal $n$-category, I'm already a bit stuck on the idea (Baez, nLab) that a commutative monoid can be defined as a monoid object in the category Mon of monoids. So what I have in a monoid object in Mon is a normal old monoid $M$ equipped with a multiplication morphism $\mu: M\times M \rightarrow M$ such that $\mu\circ(id_M\times \mu):M\times M\times M\rightarrow M= \mu\circ(\mu\times id_M): M\times M\times M \rightarrow M$ and $\mu: 1\times M\rightarrow M=\mu: M\times 1\rightarrow M=\pi_1(M\times 1)=\pi_2(1\times M)$.
In the last part the trivial monoid $1$ is identified with its image in $M$, while the formula before just says $\mu$ defines an associative binary operation on $M$. I've also identified $M\otimes M$, the monoidal category tensor, with $M\times M$. That works here, right?
All of this is separate from the multiplication on $M$ that I get just from being in Mon. Now I need somehow to connect this monoid-object-in-Mon multiplication with the original multiplication $\cdot$ on $M$ to get $a\cdot b=b\cdot a$ for each $a,b \in M$. I don't know how to do this, in part because I'm not sure how to think about that last statement categorically. Do I just define $\cdot$ as another monoidal structure on $M$ that follows the same rules as $\mu$, and try to relate them?
I appreciate any help.