Let $M$ and $N$ be normal subgroups of $G$. Find a homomorphism $f:G \rightarrow \frac{G}{M} \times \frac{G}{N}$ and use this to prove that $\frac{G}{M \cap N}$ is isomorphic to a subgroup of $\frac{G}{M} \times \frac{G}{N}$.
This was an exercise given in class which our professor handwaved the next meeting by saying ``just apply the isomorphism theorem.'' I have no idea where to start with this question as we just started our discussion on this topic.