We review Kontsevich's formulation of mirror symmetry as an equivalence of categories, and discuss the geometric properties of the (conjectural) real Fourier-Mukai functor establishing the equivalence. Specifically, we show how, in local, semi-flat, torus fibration models of a Calabi-Yau and its mirror (the dual fibration), supersymmetric A-cycles and B-cycles are related. The deformed Hermitian-Yang-Mills equations are transformed into the equations of special Lagrangian submanifolds with flat connections.