  Products in double categories, as found in cartesian double categories, are
an elegant concept with numerous applications, yet also have a few puzzling
aspects. In this paper, we revisit double-categorical products from an unbiased
perspective, following up an original idea by Par\'e to employ a
double-categorical analogue of the family construction, or free product
completion. Defined in this way, double categories with finite products are
strictly more expressive than cartesian double categories, while being governed
by a single universal property that is no more difficult to work with. We
develop the basic theory and examples of such products and, by duality, of
coproducts in double categories. As an application, we introduce finite-product
double theories, a categorification of finite-product theories that extends
recent work by Lambert and the author on cartesian double theories, and we
construct the virtual double category of models of a finite-product double
theory.
