bundle map

Let $E_{1}\\overset{\\pi_{1}}{\\to}B_{1}$ and $E_{2}\\overset{\\pi_{2}}{\\to}B_{2}$ be fiber bundles for which there is a continuous map $f:B_{1}\\to B_{2}$ of base spaces. A bundle map (or bundle morphism) is a commutative square

\\xymatrix{E_{1}\\ar[r]^{\\hat{f}}\\ar[d]_{\\pi_{1}}&E_{2}\\ar[d]^{\\pi_{2}}\\\\B_{1}\\ar[r]^{f}&B_{2}}

such that the induced map $E_{1}\\to f^{-1}E_{2}$ is a homeomorphism (here $f^{-1}E_{2}$ denotes the pullback of $f$ along the bundle projection $\\pi_{2}$ ).