Let $A$ be a set of non-empty sets, then $\bigcup A$ is a set. Furthermore, $(\bigcup A)^{A}$ is a non-empty set. Besides let $P$ be a binary predicate such that for all $X\in A$ there is a unique $x \in X$ satifies $PXx$. $f:=\{(X,x)\in A \times \bigcup A|PXx\}$ is a functional class and $f \in (\bigcup A)^A$. Hence $f$ is also a set.
According things discussed above, $f$ is indeed a choice function of $A$ and is a set. So does this fact means AC is naturally holds?