Abstract bivariant cuntz semigroups

We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups S and T, there is another Cuntz semigroup [S, T] playing the role of morphisms from S to T. Applied to C*-algebras A and B, the semigroup [[Cu(A), Cu(B)]] should be considered as the target in analogs of the universal coefficient theorem for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that orderzero maps between C*-algebras naturally define elements in the respective bivariant Cuntz semigroup.