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Is there any coherence result for (braided) monoidal functors? (like Mac Lane's coherence theorem for monoidal categories)

What I have in mind is a theorem like the following:

Let $F$ be a (braided) monoidal functor between monoidal categories $\mathsf C$ and $\mathsf D$. All compositions in the category $D$, with the same source and target, consisting of the structure maps $\alpha$, $\rho$, $\lambda$, $F(\alpha)$, $F(\rho)$, $F(\lambda)$, ($\sigma$, $F(\sigma)$) are equal.

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Yes. Have a look at:

Lewis, G. (1972) Coherence for a closed functor. In: Mac Lane, S. (ed.) Coherence in Categories. Springer-Verlag Lecture Notes in Computer Science 281, 148–195

Epstein, D. B. A. (1966) Functors between tensored categories. Invent. Math. 1, 221–228

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    $\begingroup$ These papers were published before braided categories were introduced (AFAIK, it was Joyal and Street around 1986). Do you know of any reference specifically for braided functors? Joyal and Street's 1993 paper does not have such a theorem in it. $\endgroup$ – Najib Idrissi Jul 7 '15 at 10:19
  • $\begingroup$ I'd be interested in a corresponding strictification result for symmetric monoidal functors... $\endgroup$ – David Roberts Dec 13 '15 at 13:24

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