Joint work with S. Mau (Rutgers Ph.D. candidate) and Katrin Wehrheim (MIT). We describe a realization of the multiplihedron as a moduli space of disks with additional structure, similar to the realization of the associahedra as stable marked disks. We use this realization to construct A-infinity functors for Lagrangian correspondences, and study their compositions. This project is motivated by Floer theory, which associates to a suitable three manifold and non-abelian group a homology group HF(Y). Using our results we can extend some of these invariants to three-dimensional cobordisms, taking values in A-infinity functors.