In MacLane's *Categories for the working mathematician*, the author shows that the evaluation at 1 gives an equivalence of categories $\mathrm{hom}_{\mathrm{BMC}}(B,M)\simeq M_0$ where $B$ is the braid category, $M$ is a braided monoidal category and $M_0$ is the underlying (ordinary) category to $M$. As a result of that, he states a second theorem (coherence theorem) claiming that each composite of canonical maps in $M$ induces a braiding (element of the braid group), and that two such composites are equal for all $M$ if and only they induce the same element braiding.

I understand how to construct the braiding from a given composition of canonical morphisms, but I fail to write it properly using the previous theorem and I don't know how to prove that two such maps are equal if and only if they induce the same braiding.

I also read the original preprint of Joyal and Street which appears in the references given by MacLane (http://maths.mq.edu.au/~street/JS1.pdf) but I still don't see how to write a proper proof.