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The Fourier-Mukai transform comes from a bimodule: the Poincaré bundle. Let $A$ be an abelian variety, the Poincaré bundle $\mathcal{P}$ is a vector bundle on $A \times \hat{A}$ coming from the fact that the points in the dual abelian variety $\hat{A}$ parametrize line bundles on $A$ ($\mathcal{P}$ is the universal family). In the Fourier-Mukai construction, $\mathcal{P}$ is used as a $\mathcal{O}_A$-$\mathcal{O}_{\hat{A}}$-bimodule to produce a functor between the derived categories of coherent sheaves on $A$ and $\hat{A}$ via a push-pull construction.