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