The word category is the free monoidal category on one generator

Define $T_b$ recursively on objects by $T_b(e_0)={\mathbf{1}}_{\mathcal{B}}$, $T_b((-))=b$, and $T_b(v\mathrm{\square }w)=T_b(v)\otimes T_b(w)$. If $u:v\to w$ is the unique morphism of $W$, then $v$ and $w$ have the same length, so [L4] gives a unique canonical isomorphism $T_b(v)\to T_b(w)$; define $T_b(u)$ to be that morphism. Because identities and composites in $W$ are themselves the unique arrows between equal-length words, uniqueness in [L4] makes $T_b$ a functor.

The recursive object formula already matches the tensor and unit of $W$, and the same canonical comparison maps from [L4] provide the invertible structural maps required by [L3]. Thus $T_b$ is a strong monoidal functor.

Let $S:W\to \mathcal{B}$ be another strong monoidal functor with $S((-))=b$. Build isomorphisms ${\sigma }_v:S(v)\to T_b(v)$ recursively: for $e_0$, use the inverse of the unit map of $S$; for $(-)$, use $1_b$; and for $v\mathrm{\square }w$, use the inverse of the binary structure isomorphism of $S$ followed by ${\sigma }_v\otimes {\sigma }_w$.

Since every morphism in $W$ is unique when it exists, the family ${\sigma }_v$ is automatically natural, and the same recursion forces compatibility with the monoidal structure maps. Uniqueness of the recursion makes $\sigma :S\Rightarrow T_b$ the unique monoidal natural isomorphism fixing the generator.