In general, your category has to admit small limits for that to even start to begin to make any sense at all. Also, as I noted in my question, I'm fairly sure that you've got it backwards. It should be:

Hom(colim(F(-)),X) is isomorpic to lim Hom(F(-),X), and Hom(Y,lim(F(-))) is isomorphic to lim Hom(Y,F(-)), where we're limiting and colimiting over the domain of F, where F is a functor into our category from some other category (Diagrams for example.)

I don't know if this is what you actually wanted, but if I read your question the way you typed it out, the first one doesn't make sense, since covariant hom is covariant. The second one might be true provided that the limit has certain restrictions on it or if covariant hom has an appropriate adjoint. There might be other cases, but it's not true in general. If you're just looking for the existence of a map in the second one, then it's trivial.