Let $\Phi : [\mathbb N \rightharpoondown \mathbb N] \to [\mathbb N \rightharpoondown \mathbb N]$ be the map from the set of partial functions $\mathbb N \to \mathbb N$ to itself (what's a nice way of saying this? "endomorphism" seems too strong), defined as follows:
$\Phi(f)(n) = f(n) + 1$ if $f(n)$ is defined, undefined otherwise.
I've been asked to determine whether or not this function has any fixed points. I'd like to know if the set $[ \mathbb N \rightharpoondown \mathbb N]$ includes the entirely "undefined" function, and whether this function is unique? i.e. would the function $f : \emptyset \to \emptyset$ be a fixed point of $\Phi$?