A bundle map is a map between bundles along with a compatible map between the base manifolds. Suppose 
and 
are two bundles, then
 |
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.
