Perhaps this example is too naive, but one can view the Riesz Representation Theorem categorically as saying that integration with respect to a measure is a natural equivalence of the functor which takes a compact Hausdorff space $X$ and produces the Banach space of finite signed measures on $X$, and the functor which takes and $X$ and produces the dual of $C(X)$. There is a lovely article on this by Hartig: http://www.jstor.org/pss/2975760
Another example that comes to mind is Pontryagin Duality as presented here: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102911979
