In the case of C*-algebras with unit, the spectrum map gives rise to a contravariant functor from the category of C*-algebras with unit and unit-preserving continuous *-homomorphisms, to the category of compact Hausdorff spaces and continuous maps. This functor is one half of a contravariant equivalence between these two categories (its adjoint being the functor that assigns to each compact Hausdorff space "X" the C*-algebra "C"0("X")). In particular, given compact Hausdorff spaces "X" and "Y", then "C"("X") is isomorphic to "C"("Y") (as a C*-algebra) if and only if "X" is homeomorphic to "Y".