  The category of double categories and double functors is equipped with a
symmetric closed monoidal structure. For any double category $\mathbb A$, the
corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category
$\mathbb B$ to the double category whose 0-cells are the double functors
$\mathbb A \to \mathbb B$, whose horizontal and vertical 1-cells are the
horizontal and vertical pseudotransformations, respectively, and whose 2-cells
are the modifications. Some well-known functors of practical significance are
checked to be compatible with this monoidal structure.
