I'm reading Bott&Tu's Differential Forms in Algebraic Topology and its Proposition 6.39 says that
The Euler class is functorial, i.e, if $f:N\to M$ is a $C^\infty$ map and $E$ is a rank $2$ oriented vector bundle over M, then $$ e(f^{-1}E)=f^*e(E). $$
I have no difficulty understanding this statement, but the word "functorial" is quite confusing to me. Does it mean that the Euler class can be translated into a functor between two categories? If the Euler class can be translated into a functor, then the functoriality statement should be that it commutes with composition of morphisms. However, in the equation, $e$ is applied to a vector bundles while $f^*$ is a induced map, and I just cannot find a category in which two morphisms composite to be the pullback $f^{-1}E$. This is too strange to me.
Can the Euler class really be translated into a functor? Or there is simply another meaning of "functorial" in the field of differential geometry?
Thanks in advance for any help.