I admit that the following is a very broad question. So if you feel that it is too vague please say so. It might also just be that I haven't read enough about category theory and my question is silly. If so, please point that out :-) I would also be happy if somebody just points me to some reference. If there is more to learn by replacing "abelian groups" with any other category, I am also happy.
There are many examples in mathematics where we use some functor from some category $C$ to the category of abelian groups and we can learn a lot about the original category by using our knowledge of abelian groups and certain special properties of the functor. (See for example (Co-)Homology, K-Theory) Now my question is: Can we learn anything about the objects in $C$ just from the existence of such a (non-constant) functor without knowing any details about it.
The only "obvious" use to me is the ability to classify the objects in the original category by their images.
A more reference-oriented formulation of the question: Which constraints apart from the functor being non-constant should we add to be able to infer anything interesting?
All the best, Henrik