Is Module Category over Monoidal category Monoidal?

Question

let $\mathcal{C}$ be a monoidal category and $\mathcal{M}$ a $\mathcal{C}$-module category. Does $\mathcal{M}$ need to be a monidal category? I know it is true for certain categories, but is it true in general?

Thanks,

Answer 1

Of course not. Consider the initial monoidal category $\{1\}$. Every category is then a $\{1\}$-module category. The simplest example of a category which admits no monoidal structure is the empty category.