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