  We provide an abstract categorical framework that relates the Cuntz
semigroups of the C$^*$-algebras $A$ and $A\otimes \mathcal{K}$. This is done
through a certain completion of ordered monoids by adding suprema of countable
ascending sequences. Our construction is rather explicit and we show it is
functorial and unique up to isomorphism. This approach is used in some
applications to compute the stabilized Cuntz semigroup of certain
C$^*$-algebras.
