is a bundle map if there is a map such that for all . In particular, the fiber bundle of over a point , gets mapped to the fiber of over .
In the language of category theory, the above diagram commutes. To be more precise, the induced map between fibers has to be a map in the category of the fiber. For instance, in a bundle map between vector bundles the fiber over is mapped to the fiber over by a linear transformation.
For example, when is a smooth map between smooth manifolds then is the differential, which is a bundle map between the tangent bundles. Over any point in , the tangent vectors at get mapped to tangent vectors at by the Jacobian.