I've been given a statement of the universal property of Tensor products that is as follows:
For all $A$-modules $Q$, the homomorphism $\operatorname{Hom}_A(M\otimes_A N,Q)\rightarrow \operatorname{Bil}_A(M\times N,Q)$ that sends $f\mapsto f\circ \otimes$ is an isomorphism.
I want to use this to show that a particular $\mathbb{Z}$-module $Q$ is isomorphic to a tensor product $A\otimes \mathbb{Q}$. I have managed to define a Bilinear map $B:A\times\mathbb{Q}\rightarrow Q$, but I'm not sure how to proceed from here. I would really appreciate any guidance you would be able to provide as I was unable to find any nice examples for this kind of problem online.