I don't think there is. You can for example look at the [wiki page][1] for forgetful functors to see that the functors commonly referred to as forgetful can be quite different from one another. A lot of them are fully faithful, some are just faithful and some are not even that. One hope one might have would be to define it as a right adjoint to some type of functor (some sort of "free object" functor) as discussed in [this question][2] here on MO. This has it's drawbacks since, as discussed in the Wiki page mentioned above, there are forgetful functors that do not have a left adjoint.


  [1]: http://en.wikipedia.org/wiki/Forgetful_functor
  [2]: http://mathoverflow.net/questions/6376/why-forgetful-functors-usually-have-left-adjoint