What can we learn purely from the existence of a (non-constant) functor to the category of abelian groups? 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
 A: There is a notion of a "group object" in a category (with products and a terminal object), which is an object (accompanied by a collection of nice maps) that simulates the behavior of a group. An equivalent way of stating this is that $G$ is a group object in $C$ if for every object $X \in C$ there is a group structure on $\operatorname{Hom}(X,G)$ with the assignment $X \to \operatorname{Hom}(X,G)$ being a contravariant functor $C \to \operatorname{Grp}$.
Seeing as this functor comes with a few associated conditions, it might not be the best candidate to answer your question. However, those conditions are relatively minimal, and the exact nature of this functor isn't part of the definition, i.e. what objects are sent to what groups, etc. 
Does knowing that something is a group object tell us anything meaningful? I think so. If $C = \operatorname{Diff}$, the category of smooth manifolds, then a group object is a Lie group. If a manifold is a Lie group, it is always parallelizable and orientable, and that information does not require us to know anything about the Lie group structure. There are other examples where the mere existence of some kind of group structure imposes meaningful conditions on our object of study, and this group structure can be encoded in the existence of a certain functor.
A: Just to give an example in a somewhat negative spirit, note that if $\mathcal C$ is any category equipped with a functor $U: \mathcal \to $ Sets (e.g. maybe $\mathcal C$ is the category of some sort of structures, and $U$ is the forgetful functor which just passes to the underlying set).  Given any set $S$ we may form
the free abelian group $\mathbb Z[S]$ having $S$ as a basis.  Composing,
we get the functor $X \mapsto \mathbb Z[U(X)],$ which doesn't tell us very much
about $\mathcal C$.

Note, though, that this is not such a stupid example as it might seem, nor is it as meaningless as it might seem.
E.g. suppose that $\mathcal C$ is the category of topological spaces, and $U$ is the functor $X \mapsto \pi_0(X)$ which sends a space to its set of path-connected components.  Then the composite functor considered above is precisely
$H_0$ (the $0$th homology group), and this isn't a completely worthless functor.
Of course, it is made much more interesting by the fact that it is part of the family of functors $H_i$!  

What conclusion should we draw: maybe the main one is that if, in some situation,  you can find a functor to abelian groups, then, even if the functor you find
doesn't seem very interesting, it is worth thinking about whether it (or the concepts underlying its construction) can be
enriched/generalized/enhanced to something more profound.  In the preceding example, the fairly naive notion of "connectedness" of a space can be enhanced to notions of "higher connectivity", leading to homology, homotopy, etc.
In more algebraic contexts, fairly naive constructions can often be extended to derived functors, leading to rich theories of homological algebra.
Etc. ... .
