For the endofunctor $H(X) = 1 + X$ on Set, the terminal coalgebra is $\bar{\mathbb{N}}$, the extended natural number system. Define a function $add\colon \bar{\mathbb{N}} \times \bar{\mathbb{N}} \to 1 + \bar{\mathbb{N}} \times \bar{\mathbb{N}}$:

$add(n, m) =
\begin{cases}
(pred(n), m) & if\; n \gt 0; \\
(0, pred(m)) & if\; n = 0,\; m \gt 0; \\
* & if\; m = n = 0,
\end{cases}$

Then $(\bar{\mathbb{N}} \times \bar{\mathbb{N}}, add)$ is an $H$-coalgebra. The unique coalgebra morphism ${+}\colon \bar{\mathbb{N}} \times \bar{\mathbb{N}} \to \bar{\mathbb{N}}$ (to the terminal coalgebra $\bar{\mathbb{N}}$) is addition on the extended natural numbers.