# Number of functions included in a keyed PRF

I was reading Introduction to Modern Cryptography by Katz and Lindell and i found this:

Coming back to our discussion of pseudorandom functions, recall that a pseudorandom function is a keyed function F such that $F_k$ (for $k\in2 \{0, 1\}^n$ chosen uniformly at random) is indistinguishable from $f$ (for $f\in$ $Func_n$ chosen uniformly at random). The former is chosen from a distribution over (at most) $2^n$ distinct functions, whereas the latter is chosen from all $2^{n*2n}$ functions in Funcn. Despite this, the “behavior” of these functions must look the same to any polynomial-time distinguisher.

How did he got into the conclusion that keyed function F is chosen over $2^n$ distinct functions?

Does he assume that there is a fixed function which maybe differentiated by the key k, so total of $2^n$ "functions"?

There are $2^n$ possible values for $k$, and for each one there is a function $F_k$, ergo, there are at most $2^n$ possible functions $F_k$. ("At most" is because it is possible that two different values of $k$ yield the same function $F_k$.)