  This partly expository paper first supplies the details of a method of
factoring a stable C*-algebra A as B \otimes K in a canonical way. Then it is
shown that this method can be put into a categorical framework, much like the
crossed-product dualities, and that stabilization gives rise to an equivalence
between the nondegenerate category of C*-algebras and a category of
"K-algebras". We consider this equivalence as "inverting" the stabilization
process, that is, a "destabilization".
  Furthermore, the method of factoring stable C*-algebras generalizes to
Hilbert bimodules, and an analogous category equivalence between the associated
enchilada categories is produced, giving a destabilization for
C*-correspondences.
  Finally, we make a connection with (double) crossed-product duality.
