Section of pullback bundle

Question

Suppose that $E \to B$ is a vector bundle and $f:A \to B$ is continuos. If $s$ is a section of $E$ how to define a section of pullback bundle? On wikipedia they say that it induces the section of the pullback bundle simply by defining $f^*s:=s \circ f$. Being precise, $f^*E$ is defined as $\{(x,v) \in A \times E: f(x)=p(v)\}$ where $p:E \to B$ is a projection. So an element $f^*s(x)$ should be of the form $(x,v)$ with some $x \in A$ and $v \in E$. On the other hand I read somewhere that one should use some universal property to obtain the section of the pullback bundle. So my question is

Question: how to define a pullback of a given section $s$?

Answer 1

In your characterization of the pullback as a set, you will have $f^{*}s(a) = (a, s(f(a))$. The fact that $v$ is a section ensures that $f(a) = p(s(f(a)$, as desired.

To use the universal property, you note that maps $X \to f^*E$ are equivalent to a pair of maps $g: X \to A, h: X \to E$ such that $p\circ h = f\circ g$. In your example set $X = A$, $g$ to be the identity, and $h$ to be the composite $s \circ f$. Again, $s$ being a section shows $p \circ h = p \circ s \circ f = f = f \circ g$.