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I don't think there is. You can for example look at the wiki page 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 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.