  We consider two categories of C*-algebras; in the first, the isomorphisms are
ordinary isomorphisms, and in the second, the isomorphisms are Morita
equivalences. We show how these two categories, and categories of dynamical
systems based on them, crop up in a variety of C*-algebraic contexts. We show
that Rieffel's construction of a fixed-point algebra for a proper action can be
made into functors defined on these categories, and that his Morita equivalence
then gives a natural isomorphism between these functors and crossed-product
functors. These results have interesting applications to non-abelian duality
for crossed products.
