I am trying to show that the mapping from $G \rightarrow G/H +G/K$, where + is the direct product is a homomorphism. The mapping is defined as $\phi(g)=(gH,gK)$. I have also showed that for the normal groups H and K, their intersection is also a normal subgroup of G. From that I can use that to show that $G/H\cap K$ is isomorphic to $Z\_2+Z\_2$. H and K are two distinct subgroups of G each with index 2.

Should I first show it is a homomorphism and then use to show it is isomorphic? Because their is a hint saying its tricky to show that $\phi$ is onto where $\phi$ is the homomorphism.