# Confusions over the definitions of universal bundle and characteristic class

In most references, only a principal G-bundle is called universal (ie every other bundle can be pullbacked from this one, or an equivalent definition).

Does it make sense to speak of a universal F-G bundle ? (ie a bundle with fiber F and structure group G)

Specifically, during a course, we defined the universal G-bundle in the standard fashion (Milnor construction for existence etc.). But then it was said that $EG \times_G F$ (the associated bundle) is a universal F-G bundle.

I then have the same question for characteristic classes: can they also be defined for non-principal bundles?

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If $B$ is a principal $G$-bundle, $B \times_G F$ is an $F-G$ bundle. Conversely, if $C$ is an $F -G$ bundle, $Hom (F,C)$ (structure-preserving Homs) is a principal bundle. So the two notions are equivalent.