Prasit Bhattacharya: The p-adic integers as final coalgebra

We see that the underlying set of the p-adic integers, as a metric space, can be realized as a final coalgebra as well as initial algebra of an endofunctor on pseudo metric spaces. We give two different coalgebra structure on 2-fold cartesian product of the p-adic integers such that the final map to the p-adic integers realizes the addition and the multiplication of the p-adic integers.