Predecessor on the final coalgebra (the extended natural numbers): difference of notation In the Wikipedia article, the predecessor $f$ on the final coalgebra is not defined at 0, it is only defined at $n+1$ and $\infty$.
In the $n$Lab article, $\operatorname{pred}(0) = *$, but $*$ is never defined. 
What is $*$? What is $\operatorname{pred}(0)$?
 A: Note that in the $n$Lab's notation, $\operatorname{pred}$ is a map $\bar{\mathbb{N}} \to 1 + \bar{\mathbb{N}}$, where $1$ is a one-set element and $+$ is the disjoint union. So here $*$ is simply the unique element of $1$, in other words $1 = \{*\}$. In general it's somewhat common to denote by $n$ a set of cardinality $n$ in this context.
This is because, if you recall, your functor $F$ maps a set $X$ to $1+X$; thus a coalgebra is given by a map of the form $X \to F(X) = 1+X$. The Wikipedia article, for some reason, takes the convention that an $F$-coalgebra (for this $F$) is described by a partially defined map $X \to X$. Given such a partially defined map $f : X \to X$, you can extend it to a fully defined $\tilde{f} : X \to 1+X$ by setting:
$$\tilde{f}(x) = \begin{cases}
f(x) \in X & f(x) \text{ is defined} \\
* \in 1 & f(x) \text{ is undefined}
\end{cases}$$
In the reverse direction, if $g : X \to 1 + X$ is a map, you can get a partially defined map $\underline{g} : X \to X$ by letting $\underline{g}(x) = g(x)$ if $g(x) \in X$, and leaving $\underline{g}(x)$ undefined if $g(x) = * \in 1$. These two constructions are inverse to each other, so you get a bijection between partial maps $X \to X$ and total maps $X \to 1 + X = F(X)$.
