A property of characteristic classes $c$ is that $c(f^* E) = f^* c(E)$ where $E\to M$ is a bundle and $f^* E$ is the pullback of $E$ by some map $f: N\to M$. In Bott and Tu it is stated (for Chern classes) that this implies that if $E \simeq F$ then $c(E) = c(F)$. However I am unable to see why this is true (from the naturality fact alone-- I know why its true from the construction of Chern classes). In other texts they get around this by defining characteristic classes to be defined on equivalence classes of bundles or use a more broader definition of naturality that includes bundle maps. Hence I am wondering how straightforward this statement in Bott and Tu really is.
I know from all the constructions of characteristic classes that this is a really pedantic question (obviously its isomorphism invariant!), but I am just curious.