  We study tensor structures on (Rep G)-module categories defined by actions of
a compact quantum group G on unital C*-algebras. We show that having a tensor
product which defines the module structure is equivalent to enriching the
action of G to the structure of a braided-commutative Yetter-Drinfeld algebra.
This shows that the category of braided-commutative Yetter-Drinfeld
G-C*-algebras is equivalent to the category of generating unitary tensor
functors from Rep G into C*-tensor categories. To illustrate this equivalence,
we discuss coideals of quotient type in C(G), Hopf-Galois extensions and
noncommutative Poisson boundaries.
