The proof of coherence in monoidal categories in CWM is based on the existence of a monoidal category free over a singleton. Denoting this category by $\mathcal{W}=\left(\mathcal{W}_{0},\square,e_{0},\hat{\alpha},\hat{\lambda},\hat{\rho}\right)$ it can be observed that $\mathcal{W}_{0}$ is a thin groupoid. Its objects are so-called 'binary words'. For every pair $u,v\in\mathcal{W}_{0}$ the homset $\mathcal{W}_{0}\left(u\square v,v\square u\right)$ contains exactly one arrow, and denoting it with $\hat{\gamma}_{u,v}$ it seems to me that $\left(\mathcal{W}_{0},\square,e_{0},\hat{\alpha},\hat{\lambda},\hat{\rho},\hat{\gamma}\right)$ can be recognized as a commutative monoidal category. My questions are:
1) Can $\left(\mathcal{W}_{0},\square,e_{0},\hat{\alpha},\hat{\lambda},\hat{\rho},\hat{\gamma}\right)$ be classified as a commutative monoidal category free over a singleton?
2) If the answer on the first question is 'yes' then can coherence in commutative monoidal categories be proved the same way (used in CWM) as in the proof for monoidal categories?
I think that I am overlooking complications, because the proof in CWM of coherence for monoidal categories appears to be more complex.