The problem is:
Let G and G' be groups, and let H and H' be normal subgroups of G and G', respectively. Let $\phi$ be a homomorphism of G into G'. Show that $\phi$ induces a natural homomorphism
$$\phi_*: (G/H) \rightarrow (G'/H)\ \text{if}\ \phi[H] \subseteq H'.$$ (This fact is constantly used in algebraic topology.)
Attempt at a solution:
$\phi: G \rightarrow G'$
$\phi_*(gH) = \phi(g)H'$ where $gH$ is the class of $g$ in $\frac{G}{H}$ and $\phi(g)H'$ is the class of $\phi(g)$ in $\frac{G'}{H'}$.
$$\phi_*((aH)(bH)) = \phi_*((ab)H)$$
$$ = \phi(ab)H'$$
$$= \phi(a)\phi(b)H'$$
$$= \phi(a)H'\phi(b)H'$$
$$=\phi_*(aH)\phi_*(bH)$$