What does this mean? A choice function for a set $A$ is by definition a functionthe choice function across the ordinal indexes $i$ for each countable $s_i$
$$f:A\rightarrow \bigcup A$$
such that $f(x)\in x$. Is this what you mean? If so, what is the $A$ here?
If not, what do you mean with "the choice function across ordinal indexes $i$ for each countable $s_i$"? Define it please.