Is $D$ symmetric monoidal?

We verify that a certain functor $D\text{\hspace{0.17em}⁣}:{\text{Sp}}^{\mathrm{\Sigma }}({\text{Ch}}^{+})\to \text{Ch}$ is symmetric monoidal. This functor is used elsewhere in developing the model category theory of symmetric spectra and of chain complexes graded over $\mathbb{N}$ or $\mathbb{Z}$.