Correct formulation of axiom of choice In a paper (Wayback Machine), Asaf Karagila writes:

Definition 1 (The Axiom of Choice). If $\{A_i
\mid i ∈ I\}$ is a set of non-empty sets, then there exists
  a function $f$ with domain $I$ such that $f(i) ∈ A_i$
  for all $i ∈ I$.

Does this formally make sense? Shouldn't it say

If $(A_i)_{i\in I}$ be a family of non-empty sets …

?
Because if we have a set $M$ of non-empty sets there may be no unique family $(A_i)_{i\in I}$ with $\{A_i\mid i\in I\}= M$.

It does make sense subject to a generous interpretation of the role of the indexing set $I$ and the indexing function $i \mapsto A_i$. It would be much better style (in my opinion) either to write it as you suggested, stating explicitly that the indexing function $i \mapsto A_i$ is part of the data or to write it without mentioning the indexing set at all: "if $U$ is a set of non-empty sets, then there is a function $f$ with domain $U$ such that $f(A) \in A$ for every $A \in U$". The two statements are equivalent (because any indexing function $i \mapsto A_i$ of the set $U$ factors through the identity function on $U$, which you can regard as a sort of "minimalist" indexing function).
[Aside: I don't think there is any general agreement that "family" means indexed family. Indeed, some authors explicitly state that they use terms like "set", "family" and "collection" as synonyms. So it's safest to say "indexed set" or "indexed family".]
