I'll report on joint work with Nick Sheridan (Princeton/IAS) about mirror symmetry for Calabi-Yau (CY) manifolds. Kontsevich's homological mirror symmetry (HMS) conjecture proposes that the Fukaya category of a CY manifold (viewed as a symplectic manifold) is equivalent to the derived category of coherent sheaves on its mirror. We show that if one can prove an apparently weaker fragment of this conjecture, for some mirror pair, then one can deduce HMS for that pair. We expect this fragment  to be amenable to proof for the mirror pairs constructed in the Gross-Siebert program, for example.  We also show that the "closed-open string map" is an isomorphism, thereby opening a channel for proving the "closed string" predictions of mirror symmetry.