I'm trying to prove that localization commutes with tensor products (Dummit and Foote. Abstract algebra, Sec 15.4 Ex 16). There is a unique isomorphism of $D^{-1}R$-modules
$$\varphi: D^{-1}M\otimes_{D^{-1}R}(D^{-1}N)\cong D^{-1}(M\otimes_{R}N)$$
with $\varphi(\frac{m}{d}\otimes \frac{n}{d'})$ given by $\frac{m\otimes n}{dd'}$ for any $R$-modules $M,N$, and multiplicatively closed set $D$ in $R$. {I think I have managed to prove the existence of isomorpism as follows: $$(D^{-1}M)\otimes_{D^{-1}R}(D^{-1}N)\cong(D^{-1}R\otimes_{R}M)\otimes_{D^{-1}R}(D^{-1}R\otimes_RN)\\$$ $$\cong D^{-1}R\otimes_R(M\otimes_{D^{-1}R}D^{-1}R)\otimes_{R}N\cong D^{-1}R\otimes_{R}M\otimes_{R}N\cong D^{-1}(M\otimes_{R}N)$$
However, I am stuck in uniqueness part. Could anyone help? Thanks.