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.