ion elimination Definition 3.1. A monoidal category where every object has a commutative comonoid structure is said to be semi-cartesian. An action category is a K\Omega -category with a distinguished admissible commutative comonoid structure on every object. A semi-cartesian category is cartesian if and only if each object carries a unique comonoid structure, and such structures form two natural families, \Delta and !. The naturality means that all morphisms of the category must be comonoid homomorphisms. In action categories, the property of semi-cartesianness is fixed as structure: on each object, a particular comonoid structure is chosen. This choice may be constrained by some given graphic operations, with respect to which the structures must be admissible. The proof of proposition 2.6 shows that such structures determine the abstraction operators, and are determined by them. This is the essence of the equivalence of action categories and action calculi. As the embodiment of 2...