  We previously showed that abstract Cuntz semigroups form a closed symmetric
monoidal category. This automatically provides additional structure in the
category, such as a composition and an external tensor product, for which we
give concrete constructions in order to be used in applications.
  We further analyse the structure of not necessarily commutative Cu-semi-rings
and we obtain, under mild conditions, a new characterization of solid
Cu-semirings $R$ by the condition that $R\cong [\![ R,R ]\!]$.
