  We show that any Abelian module category over the (degenerate or quantum)
Heisenberg category satisfying suitable finiteness conditions may be viewed as
a 2-representation over a corresponding Kac-Moody 2-category (and vice versa).
This gives a way to construct Kac-Moody actions in many
representation-theoretic examples which is independent of Rouquier's original
approach via `control by K_0.' As an application, we prove an isomorphism
theorem for generalized cyclotomic quotients of these categories, extending the
known isomorphism between cyclotomic quotients of type A affine Hecke algebras
and quiver Hecke algebras.
