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. From this definition, we may read off these basic facts about $+$:

$n + m \gt 0$ with $pred(n + m) = pred(n) + m$ if $n \gt 0$,

$n + m \gt 0$ with $pred(n + m) = 0 + pred(m)$ if $n = 0$ and $m \gt 0$,

$n + m = 0$ if $n = 0$ and $m = 0$.

(From the last two, it’s immediate to prove by coinduction that $0 + m = m$ for all $m$.)