The $p$-adic integers as final coalgebra

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