Let $S$ be any non-empty set, an indexing is a function $f:\mathcal{I} \rightarrow S$ such that $f$ is surjective. It is not hard to look for $f$ and $\mathcal{I}$ satisfying this definition (take the identity function). But let's say I want an indexing that allows for repetitions, that is, surjective $f:\mathcal{I} \rightarrow S$ such that for every $x\in S$ $|f^{-1}(\{x\})| \in \mathbb{N}$, how should I construct such an indexing? In my head, this seems doable, but I'm worried there are some set theoretic facts that I am overlooking.
Edit: I think I need the Axiom of Choice to actually construct this. Am I right?