Proposition 1.5. Given a map $f\colon A\to B$ and a vector bundle $p\colon E\to B$, then there exists a vector bundle $p′\colon E′\to A$ with a map $f′\colon E′\to E$ taking the fiber of $E′$ over each point $a\in A$ isomorphically onto the fiber of $E$ over $f(a)$, and such a vector bundle $E′$ is unique up to isomorphism
Often the vector bundle $E′$ is written as $f^∗(E)$ and called the bundle induced by $f$, or the pullback of $E$ by $f$.
(Vector Bundles and K-Theory, Allen Hatcher)
Is $f′$ a vector bundle homomorphism from $f^∗(E)$ to $E$?