  We show that a pair of complementary dagger-Frobenius algebras, equipped with
a self-conjugate comonoid homomorphism onto one of the algebras, produce a
nontrivial unitary morphism on the product of the algebras. This gives an
abstract understanding of the structure of an oracle in a quantum computation,
and we apply this understanding to develop a new algorithm for the
deterministic identification of group homomorphisms into abelian groups. We
also discuss an application to the categorical theory of signal-flow networks.
