What is a endomorphism of vector bundle? Quick question: When we say $f:E\to E$ is an endomorphism of the vector bundle $\pi:E\to M$, do we require that $f$ maps each fiber $E_p$ to itself, or it could be to another fiber $E_q$?
I couldn't find the answer online. Any reference?

Yes, $f$ must map $E_p$ to itself. Here is how Clifford Taubes defines it in his recent book Differential Geometry: Bundles, Connections, Metrics and Curvature:

Let $\pi: E\to M$ and $\pi': E'\to M$ denote a pair of vector bundles. A homomorphism
  $\mathfrak{h}: E \to E'$ is a smooth map with the property that if $p \in M$ is any given point, then $\mathfrak{h}$ restricts to $E|_p$ as a linear map to $E'|_p$. An endomorphism of a given bundle $E$ is a homomorphism from $E$ to itself.

