In [CWM, Ch. XI, §1], Mac Lane prove the coherence theorem for symmetric monoidal categories by assuming the strictness. Thus we have a $n-$ary tensor functor $T$, and the theorem states that any two chains of braiding isomorphisms connecting two permutations of $T$ must be equal.
Mac Lane claims that any closed chain corresponds to a relation of the generators of the $n$th symmetric group. But how?
It seems that the corresponding is given by sent a braiding isomorphism to a transposition. For example, let $12\cdots n$ denote the functor $X_1,X_2,\cdots,X_n\mapsto X_1\otimes X_2\otimes\cdots\otimes X_n$ and others likewise. Then the braiding isomorphism $12\cdots n\to21\cdots n$ should be corresponded to $(12)$. But, for example, $23145\cdots n\to32145\cdots n$ should also be corresponded to $(12)$. Note that every functor can be obtained by applying a permutation on $12\cdots n$, thus we can corresponding them to the permutation. So a braiding isomorphism should correspond to the quotient of the permutations corresponded to its domain and codomain.
But then every closed chain would produce the trivial relation, which is not as desired.
So, what's wrong?