Let $\xi = \pi : E \rightarrow X$ be a vector bundle and $f: Y \rightarrow X$ a continuous map.
Let $E' \subset Y \times E$ be the set of all $(y,e)$ with $f(y) = \pi(e)$. We can define a vector bundle $f^*(\xi)= \pi' : E' \rightarrow Y$ with $\pi'(y,e)=y$ and $\pi'^{-1}(y)=\{ (y,e)| e\in \pi^{-1}(f(y)\}$.
Also $\overline f: E' \rightarrow E$ $\quad$ $\overline f (y,e)=e$ induces a bundle map $(\overline f, f)$ which is an isomorphism on each fibre.
I need to show that if $\xi$ is orientable then $f^*(\xi)$ is also orientable but i don't know how to check the compatibility condition in the definition of orientability.