Reedy diagrams in symmetric monoidal model categories

Given a small category $I$ and a closed symmetric monoidal category $\text{mm}$, we show that the diagram category ${\text{mm}}^I$ with the objectwise product is a closed symmetric monoidal category. We then prove that if $I$ is a Reedy category and $\text{mm}$ has a model structure compatible with its product, then so is the Reedy model structure on ${\text{mm}}^I$ provided that $\text{mm}$ is cofibrantly generated.