The p-adic Integers as Final Coalgebra

We express the classical p-adic integers \(\hat{\mathbb {Z}}_p\), as a metric space, as the final coalgebra to a certain endofunctor. We realize the addition and the multiplication on \(\hat{\mathbb {Z}}_p\) as the coalgebra maps from \(\hat{\mathbb {Z}}_p\times \hat{\mathbb {Z}}_p\).