  Monoidal categories enriched in a braided monoidal category $\mathcal{V}$ are
classified by braided oplax monoidal functors from $\mathcal{V}$ to the
Drinfeld centers of ordinary monoidal categories. In this article, we prove
that this classifying functor is strongly monoidal if and only if the original
$\mathcal{V}$-monoidal category is tensored over $\mathcal{V}$. We then define
a completion operation which produces a tensored $\mathcal{V}$-monoidal
category $\overline{\mathcal{C}}$ from an arbitrary $\mathcal{V}$-monoidal
category $\mathcal{C}$, and we determine many equivalent conditions which imply
$\mathcal{C}$ and $\overline{\mathcal{C}}$ are $\mathcal{V}$-monoidally
equivalent.
  Since being tensored is a property of the underlying $\mathcal{V}$-category
of a $\mathcal{V}$-monoidal category, we begin by studying the equivalence
between (tensored) $\mathcal{V}$-categories and oplax (strong)
$\mathcal{V}$-module categories respectively. We then define the completion
operation for $\mathcal{V}$-categories, and adapt these results to the
$\mathcal{V}$-monoidal setting.
