Word normalization proof of monoidal coherence

Associate to a finite ordered word $w$ the right-associated tensor $N(w)$, retaining a terminal unit: $N(\varnothing )=I$ and $N(Aw)=A\otimes N(w)$. Define concatenation maps recursively byThe unit identities derived from the monoidal pentagon and triangle give $c_{u,\varnothing }={\rho }_{N(u)}$. Induction on $u$, using the pentagon in the induction step and left-unitor compatibility in the base case, givesNormalize a letter by ${\rho }_A^{-1}$, a formal unit by $1_I$, and a tensor expression recursively using $c$. The displayed identities show that normalization commutes with each generating associator or unitor, and therefore also with inverses, tensor products and composites of them. Every structural arrow from expression $S$ to expression $T$ is consequently ${\nu }_T^{-1}{\nu }_S$, proving monoidal coherence without presupposing it.