In a cartesian multicategory, the cartesian product has a “mapping out” universal property like you would expect for a positive type: Hom(A x B ; C) = Hom(A,B ; C). The fact that such an object is equivalent to a categorical cartesian product (with usual its negative “mapping in” universal property) depends on cartesianness of the multicategory, which in turn encodes the structural rules of weakening and contraction in type theory. Thus the category theory exactly mirrors the type-theoretic fact that in linear type theory, the positive and negative products are different.

P ≃ ∀α. (P → α) → α

If P is positive, then this asserts the equivalence of a positive and a negative type! This is quite a remarkable fact, the more so since it’s true. :)

From the categorical point of view, this reflects the insufficiently-appreciated fact that abstract types are neither algebras nor coalgebras. The categorical approach to (co)algebra is inherently functorial, and since APIs frequently and naturally are mixed-variance, you can’t really define functorial actions for them. You really need parametricity to explain this, and its categorical formulation is nontrivial, to say the least.

None of this is a knock on polarity, of course — it’s really amazing just how much mileage you can get out of it, and I don’t know anything in PL theory with a better power-to-weight ratio.