Given two sets $A$ and $B$ and a function $F$ from $B$ to multisets of $A$, e.g.:
\begin{align} A &= \{x, y, z\} \\ B &= \{b_1, b_2, b_3\} \\ F(b_1) &= \{x,x,y\}\\ F(b_2) &= \{y,z,z\}\\ F(b_3) &= \varnothing\\ \end{align}
Is the following function notation for $F$ correct?
\begin{align} F &: B \to (A \to \mathbb{N)} \end{align}