The bordered Floer homology of a 3-manifold with torus boundary can be interpreted  as a collection of immersed curves in the punctured torus. Similarly, the bordered Floer homology of a 3-dimensional cobordism M:T^2->T^2  give an operator F_M which eats an immersed curve on the punctured torus and spits out another such curve. (More formally, F_M is a functor from the Fukaya category of the punctured torus to itself.) I'll explain how to describe some interesting examples (the cabling operator, for example) using correspondences.  Based on joint work with Holt Bodish and James Pascaleff.