On the tensor product of completely distributive quantale-enriched categories

Tensor products are ubiquitous in algebra, topology, logic and category theory. The present paper explores the monoidal structure of the category $\mathcal{V}\hspace{0pt}\mbox{-}\hspace{.5pt}\mathbf{Sup}$ of separated cocomplete enriched categories over a commutative quantale $\mathcal{V}$, the many-valued analogue of complete sup-lattices. We recover the known result that $\mathcal{V}\hspace{0pt}\mbox{-}\hspace{.5pt}\mathbf{Sup}$ is $*$-autonomous and we show that the nuclear/dualizable objects in $\mathcal{V}\hspace{0pt}\mbox{-}\hspace{.5pt}\mathbf{Sup}$ are precisely the completely distributive cocomplete $\mathcal{V}$-categories.