Let me expand on the second example of Lian. Category theory proves trivially that the category of profinite abelian groups is dual to the category of discrete torsion abelian groups. Indeed the former is the pro-completion of the category of finite abelian groups and the latter is the ind-completion. So one just needs the trivial fact that the category of finite abelian groups is self-dual.

Similarly, assuming the Peter-Weyl theorem (you can't get around this) one has that the category of compact abelan groups is the pro-completion of the category of compact abelian Lie groups. The category of discrete abelian groups is the ind-completion of the category of finitely generated abelian groups. So the duality between compact abelian groups and discrete abelian groups boils down to the structure theorem for finitely generated abelian groups, the finite case and that Z is dual to the circle.

So category theory organizes these proofs (the proof still boils down to the same nuts and bolts).