Proving associativity in monoidal category: Free Monoid construction. I am filling in the details of Mac Lane's proof of the following: If monoidal category $B$ has countable coproducts, and if the functors $-\square  a$ and $a\square -$ preserve them, then the evident forgetful functor $U$ from $\text{Mon}_{B}$ to the underlying category $B$ has a left adjoint. The proof is straightforward enough, except that I can't seem to verify that $\mu (1\square \mu )=\mu (\mu \square 1)\alpha $.
The functor $F$ takes each $a\in B$ to $\coprod _{n}a^{n}$, and $\mu $ is defined by juxtaposition; formally, by producing a unique arrow $\phi :\coprod _{m,n}a^{m}\square a^{n}\rightarrow \coprod _{k}a^{k}$, from the UMP of the coproduct. I have been able to show associativity in $Set$, but I can't seem to produce the right diagram to show this in general. Any hints or suggestions would be greatly appreciated. It's my first pass at the subject and I want to dot the i's and cross the t's. 
 A: If I understand correctly, you want to show that the outer pentagon of the following diagram commutes.

The middle square commutes because the arrows going upwards are, on the individual summand $a^k\otimes(a^l\otimes a^m)$, the tensor product of the inclusions $i_k:a^k\to\coprod a^k,\ i_l,\ i_m$, so this is just naturality of $\alpha$.
Zooming into the diagram, the right hand square looks like this

Here every triangle and square commutes. In a similar manner, one can show that the left hand square of the first diagram commutes. It follows that $\mu(\mu\otimes1)\alpha = \mu(1\otimes\mu)$.
A: Your first map
$$\coprod_{k,m,n} a^k\otimes(a^m\otimes a^n)\to Fb\otimes(\coprod_{m,n} a^m\otimes a^n)\to Fb\otimes (Fb\otimes Fb)\to^{\mu(1\otimes \mu)} Fb$$
is given on components $a^k\otimes (a^m\otimes a^n)$ by the composition of the associators $a^k\otimes(a^m\otimes a^n)\to a^k\otimes a^{m+n}\to a^{k+m+n}$. The other map is given by the composition of associators $a^k\otimes(a^m\otimes a^n)\to (a^k\otimes a^m)\otimes a^n\to a^{k+m}\otimes a^n\to a^{k+m+n}$. So this is just the coherence theorem applied to iterates of $\alpha$.
