Hi Cheuk, don't think that analogy is going to help. You can look at [this](https://math.stackexchange.com/questions/1009615/connection-between-categorical-notion-of-adjunction-and-dual-space-adjoint-in-ve) math.SE quetion, [this](http://dx.doi.org/10.1090/S0002-9939-1974-0346548-4) paper, and in ch. 3 of this [book](https://www.researchgate.net/publication/235541556_Categorical_quantum_models_and_logics) of Chris Heunen there is a nice diagram of the wider situation. So you can complicate the problem in terms of enriching, or follow the correspondence from dual maps to certain Galois connections, but it is irreversible.
If you want an example of naturality in that area, in Mac Lane's book he explains how **Vect** is not naturally isomorphic with its dual, but it is, with its **double** dual.
EDIT: for finite dimensional spaces