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.