I'm having trouble proving exercise 6.11.3 of "Introduction to homological algebra" by Weibel. I need to show that the category of torsion abelian groups is dual to the category of profinite abelian groups. It also gives a hint to show that $A$ is a torsion abelian group iff $\hom(A,\mathbb{Q}/\mathbb{Z})$ is a profinite group.
I'm stuck with the hint. I've proved that the torsion abelian group part implies that $$\hom(A,\mathbb{Q}/\mathbb{Z}) = \lim_{\leftarrow} \hom(H,\mathbb{Q}/\mathbb{Z}),$$ with $H$ going through all finite subgroups of $A$ with restriction maps as homomorphisms in the obvious way. I have absolutely no idea how to proof the other implication. I also don't see how this is going to help to associate a torsion abelian group to a profinite abelian group to make the duality.
Any thoughts ?