Canonical morphisms between parenthesised tensor words

Let $u$ and $v$ be parenthesised tensor words on the same ordered letters $x_1,\dots ,x_n$ (Parenthesised tensor words and their evaluation functors) and let $E_u,E_v:{\mathcal{C}}^n\to \mathcal{C}$ be their evaluation functors in a monoidal category $\mathcal{C}$ (Monoidal category).

A canonical morphism from $u$ to $v$ is a natural transformation $E_u\Rightarrow E_v$ (Natural transformation and its components) belonging to the smallest class closed under:

• identities $1_{E_u}$;
• the associator, left unitor, and right unitor of $\mathcal{C}$, together with their inverses, whenever their source and target are evaluation functors of parenthesised words;
• tensoring a canonical morphism with an identity natural transformation on the left or right, whenever the resulting source and target still come from parenthesised words;
• vertical composition of canonical morphisms.

Thus a canonical morphism is built only from the structural isomorphisms $\alpha ,\lambda ,\rho$, their inverses, identities, tensoring with identities, and composition. No arbitrary morphism of $\mathcal{C}$ is canonical merely because its source and target happen to be tensor products of the same objects.