Communication between the product and coproduct families requires the addition of interaction principles to the (dual) residuation laws. The principles above take the form of inference rules obtained from the following recipe: from A ⊗ B ⊢ C ⊕ D in the premise, one selects a product and a coproduct term; in the conclusion, one simultaneously introduces the residual operations for the remaining two terms. Using the (dual) residuation laws, one derives the following patterns from the interaction principles. Alternatively, one can take (P1)–(P4) as primitive postulates, and obtain the interaction principles as derived inference rules, using transitivity and the (dual) residuation laws.