SCHWEITZER 1997 Problem 6

Let $\kappa$ be an infinite cardinality and let A , B be sets of cardinality $\kappa$ . Construct a family $\cal F$ of functions $f : A \to B$ with cardinality $2^\kappa$ such that for all functions $f_1,\cdots, f_n \in\cal F$ and for all $b_1 , ..., b_n \in B$ , there exist $a\in A$ such that $f_1(a) = b_1,\cdots, f_n(a) = b_n$ .