If $\pi:E\to B$ is a bundle and $f:B^{{\prime}}\to B$ is an arbitrary continuous map, then there exists a pullback, or induced, bundle $f^{*}(\pi):E^{{\prime}}\to B^{{\prime}}$, where

and $f^{*}(\pi)$ is the restriction of the projection map to $B^{{\prime}}$. There is a natural bundle map from $f^{*}(\pi)$ to $\pi$ with the map $B^{{\prime}}\to B$ given by $f$, and the map $\varphi:E^{{\prime}}\to E$ given by the restriction of projection.

If $\pi$ is locally trivial, a principal $G$-bundle, or a fiber bundle, then $f^{*}(\pi)$ is as well. The pullback satisfies the following universal property:

(i.e. given a diagram with the solid arrows, a map satisfying the dashed arrow exists).