Thus f*E is the disjoint union of all these fibers equipped with a suitable topology.
Define the pullback bundle by and equip it with the subspace topology and the projection map π′ : f*E → B′ given by the projection onto the first factor, i.e., The projection onto the second factor gives a map such that the following diagram commutes: If (U, φ) is a local trivialization of E then (f−1U, ψ) is a local trivialization of f*E where It then follows that f*E is a fiber bundle over B′ with fiber F. The bundle f*E is called the pullback of E by f or the bundle induced by f. The map h is then a bundle morphism covering f. Any section s of E over B induces a section of f*E, called the pullback section f*s, simply by defining If the bundle E → B has structure group G with transition functions tij (with respect to a family of local trivializations {(Ui, φi)}) then the pullback bundle f*E also has structure group G. The transition functions in f*E are given by If E → B is a vector bundle or principal bundle then so is the pullback f*E. In the case of a principal bundle the right action of G on f*E is given by It then follows that the map h covering f is equivariant and so defines a morphism of principal bundles.
The pullback of bundles then corresponds to the inverse image of sheaves, which is a contravariant functor.
The tension and interplay between bundles and sheaves, or inverse and direct image, can be advantageous in many areas of geometry.
However, the direct image of a sheaf of sections of a bundle is not in general the sheaf of sections of some direct image bundle, so that although the notion of a 'pushforward of a bundle' is defined in some contexts (for example, the pushforward by a diffeomorphism), in general it is better understood in the category of sheaves, because the objects it creates cannot in general be bundles.