  Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic
Calabi-Yau three-fold $X$ is seen to have an expected adiabatic form for its
induced cohomology operation only when an appropriately twisted operation resp.
twisted charge is defined. Some differences with the case of $K3$ as well as
connections with the spectral cover construction for bundles on $X$ are pointed
out. In the context of mirror symmetry Kontsevich's association of line bundle
twists (resp. a certain 'diagonal' operation) with monodromies (esp. the
conifold monodromy) is made explicit and checked for two example models.
Interpreting this association as a relation between FM transforms and
monodromies, we express the fibrewise FM transform through known monodromies.
The operation of fibrewise duality as well as the notion of a certain index
relevant to the computation of the moduli space of the bundle is transported to
the sLag side. Finally the moduli space for D4-branes and its behaviour under
the FM transform is considered with an application to the spectral cover.
