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?