I will describe some results on functoriality of Gromov-Witten invariants under symplectic quotients. Mirror theorems in the sense of Givental relate genus zero gauged Gromov-Witten theory of a G-variety with the Gromov-Witten theory of the symplectic quotient, for the case of semipositive toric varieties. A geometric approach to the relation in general was initiated by Gaio-Salamon and Ziltener. I will describe a notion of morphism of CohFT based on the complexification of Stasheff's multiplihedron that provides an algebraic home for the bubbling considered by Gaio-Salamon. Our proposal is that a "quantum Kirwan" morphism of CohFT's extends Givental's "mirror map" to arbitrary X,G