Isomorphism from module $A\times B\otimes_{A\times B} M$ to product module $A\otimes_A M\times B\otimes_B M$.

Question

Given two rings, $A$, $B$, and a module $(A\times B)\otimes M$, is there an isomorphism $$\big(A\otimes_A M\big)\times \big(B\otimes_B M\big) \overset{h}{\longrightarrow} \big(A\times B\big) \otimes_{A\times B} M$$

I've tried some of the (seemingly) obvious choices, but none of them seem to work out. If there is one, a push in the right direction would be much appreciated.