Mac Lane coherence in canonical-map form

For a consecutive block $x_i,\dots ,x_{i+m-1}$, let $r_{i,0}:=\mathbf{1}$, $r_{i,1}:=x_i$, and, for $m\ge 1$, let $r_{i,m+1}:=(r_{i,m}\otimes x_{i+m})$. We claim that every parenthesised tensor word $w$ whose ordered letters are exactly this block admits a canonical natural isomorphism ${\kappa }_w:E_w\Rightarrow E_{r_{i,m}}$.

The claim is proved recursively. For a word containing no letters, repeated unitors give a canonical map to $r_{i,0}=\mathbf{1}$; for a one-letter word, repeated unitors give a canonical map to that actual letter $r_{i,1}=x_i$. For $w=(a\otimes b)$, suppose $a$ contains the first $p$ letters $x_i,\dots ,x_{i+p-1}$ and $b$ contains the next $q$ letters $x_{i+p},\dots ,x_{i+p+q-1}$. Recursion gives ${\kappa }_a:E_a\Rightarrow E_{r_{i,p}}$ and ${\kappa }_b:E_b\Rightarrow E_{r_{i+p,q}}$. Their tensor is canonical, and repeated associators and unitors give a canonical isomorphism $E_{r_{i,p}\otimes r_{i+p,q}}\Rightarrow E_{r_{i,p+q}}$. The composite is ${\kappa }_w$, and every step uses only the generators allowed in [L1].

For the given words $u$ and $v$, define ${\theta }_{u,v}:={\kappa }_v^{-1}\circ {\kappa }_u:E_u\Rightarrow E_v$. This is a canonical natural isomorphism.

By [L2], the ambient monoidal category is equivalent to a strict one, so [L3] applies. Hence any two canonical morphisms from $u$ to $v$ are equal. Since step 3.1 produced one such morphism, it is the unique canonical natural isomorphism $E_u\Rightarrow E_v$.