  We prove that the 2-Deligne tensor product of two compact semisimple
2-categories exists. Further, under suitable hypotheses, we explain how to
describe the $Hom$-categories, connected components, and simple objects of a
2-Deligne tensor product. Finally, we prove that the 2-Deligne tensor product
of two compact semisimple tensor 2-categories is a compact semisimple tensor
2-category.
