Is the following a restatement of the axiom of choice?

Given an arbitrary collection $A$ of non empty sets there exists a set $C$ which contains precisely one element from each element of $A$.

According to Munkres, given a collection $\mathbb{B}$ of non empty sets there exists a choice function so that $c(B) \in B$ for each $B \in \mathbb{B}$. So why can’t I define my set to be the set containing $c(B)$ for $B \in \mathbb{B}$?