  We show how the notion of intercategory encompasses a wide variety of
three-dimensional structures from the literature, notably duoidal categories,
monoidal double categories, cubical bicategories, double bicategories and Gray
categories. Variations on the notion of span provide further examples of
interest, an important one being the intercategory of sets. We consider the
three kinds of morphism of intercategory as well as the cells binding them with
applications to the above structures. In particular hom functors are studied.
