# Section of pullback bundle

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 $$s$$?

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 $p \circ h = p \circ s \circ f = f = f \circ g$.