$\array{
F A \otimes F B &\stackrel{\sigma}{\to}& F B \otimes F A
\\
{}^{\mathllap{\nabla_{A,B}}}\downarrow
&&
\downarrow^{\mathrlap{\nabla_{B,A}}}
\\
F(A\otimes B) &\stackrel{F(\sigma)}{\to}& F(B \otimes A)
}$

commutes, where $\sigma$ denotes the symmetry isomorphism both of $C$ and $D$.

Properties

As long as it goes between symmetric monoidal categories a symmetric monoidal functor is the same as a braided monoidal functor.