Are there examples where one proves something about the functor represented by an object using the functor it corepresents? Are there any interesting examples where one proves something about a representable functor $\mathrm{Hom}(-,X)$ by using the functor $\mathrm{Hom}(X,-)$?
By Yoneda's lemma, these two functors contain the same information as $X$ itself, so anything about one can be expressed in some uninteresting way as a property of the other. (For example, a nonempty topological space $X$ is connected if and only if every map from $X$ to the two-point discrete set factors through one of the points. This is best expressed in terms of the functor $\mathrm{Hom}(X,-)$, but using Yoneda's lemma, you could also do it in a silly way in terms of the functor $\mathrm{Hom}(-,X)$.) I'm not interested in these examples, but to rule them out, I'd have to know a way of formalizing the vague concept of Yoneda property, which I don't. I want genuine examples where one proves something most naturally expressed in terms of maps into $X$ by using things which are most naturally expressed in terms of maps out of $X$.
This question was motivated by discussions in the comments here and here.
 A: An absolute Galois group is an inverse limit of finite Galois groups over a system of finite Galois extensions of fields, so it represents a functor on groups defined by a compatible system of homomorphisms.  As you no doubt know, many mathematicians like to describe Galois representations, i.e., maps from such a group to groups of linear transformations, and such information arises as part of the functor the group corepresents.  I think this provides a good collection of examples, since careful study can produce interesting information concerning the Galois groups over our base field (together with large chunks of number theory and arithmetic geometry).
A: This is rather a (long) comment.
I don't think that something like this exists or is at least useful. The only chance could be if the category has an anti-autoequivalence (e.g. finite abelian groups, $A \mapsto Hom(A,S^1)$). I want to comment on

By Yoneda's lemma, these two functors contain the same information as X  itself

This is not true. For a covariant functor $F$, morphisms $Hom(X,-) \to F$ correspond to elements of $F(X)$, and for a contravariant functor $G$, morphisms $Hom(-,X) \to G$ correspond to elements of $G(X)$. But what about morphisms in the other direction? I think that these hom-functors, regarded as objects in the functor category, contain much more information than $X$, and they are not related at all. Of course, you could restrict yourself to the category of representable functors, but then somehow it is artificial to talk about these functors, right?
I think it would be the best if you give us at least one example?
A: An endomorphism of a finite set (or finite-dimensional vector space) is an injection if and only if it is a surjection.
For sets and vector spaces, injections are the same as monomorphisms, and surjections are the same as epimorphisms. So these are very naturally representable/co-representable properties. (A map $X\to Y$ is said to be a monomorphism if the maps $\mathrm{Hom}(Z,X)\to\mathrm{Hom}(Z,Y)$ are injective for all $Z$, and it's an epimorphism if the maps $\mathrm{Hom}(X,Z)\to\mathrm{Hom}(Y,Z)$ are all injections.) 
NB This isn't technically an answer to the question, since these are properties of morphisms and not of objects, but it has the right spirit.
