Wikipedia's article on presentations of groups says: If $S$ is indexed by a set I consisting of all the natural numbers $\mathbb{N}$ or a finite subset of them, then it is easy to set up a simple one to one coding (or Gödel numbering) $f : F_S → \mathbb{N}$ from the free group on $S$ to the natural numbers, such that we can find algorithms that, given $f(w)$, calculate $w$, and vice versa.
What exactly is meant here? What I'm able to get from that is the existence of an injective function $f \colon F_S \to \mathbb{N}$ such that $Im(f)$ is recursively enumerable and $g = f \circ \eta_S$, where $g$ is the Gödel numbering $S \to \mathbb{N}$ and $\eta$ is the counit of the adjoint pair. Or is there something else going on?
There's not much going on here, I would not make too much of the phraseology "Gödel numbering". Elements of $F_S$ are finite sequences of symbols in the alphabet $S \cup \bar S = \bigcup{s \in S} \{s,\bar s\}$ such that the sequence is reduced meaning that there is no consecutive subsequence of the form $s \bar s$ or $\bar s s$. All this is saying is that there is a computer program which writes a list of all the reduced words without repetition, where the set $S$ itself is expressed by enumerating it with natural numbers; I suppose one must also decide how one wants to express the formal inverses $\bar S$.