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 $F_k$.)