Suppose $\pi$ is the projection, $E$ total space, $B$ the base space, and $F$ the fiber.
A section of a fiber bundle is a continuous map $f\colon B \to E$ such that $\pi(f(x))=x$ for all $x \in B$.
Suppose that $M$ and $N$ are base spaces, and $\pi_E : E \to M$ and $\pi_F : F \to N$ are fiber bundles over $M$ and $N$, respectively. A bundle map consists of a pair of continuous functions
$\varphi\colon E\to F,\quad f\colon M\to N$
such that $\pi_F\circ \varphi = f\circ\pi_E$.
I wonder how the following is consistent with the definition of section above?
A bundle map from the base space itself (with the identity mapping as projection) to $E$ is called a section of $E$.
Specifically, how is the other fiber bundle like for the bundle map?
Added: I wonder if people think of a section more often as a "inverse" of projection, or as a bundle map? What is the purpose of viewing a section as a bundle map, which seems to me so indirect?
All quotes are from Wikipedia.
Thanks and regards!