simplifying "$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}$"

Question

Is this equality correct?

For finite sets $A$ and $B_a$ (where $a\in A$), we have: $$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}=\sum_{f\in \prod_{a\in A}B_a}\quad \prod_{a\in A}{h(a,f(a))}$$

Answer 1

Yes, this is correct. The right-hand side is the result of applying the distributive law to the left-hand side.