I am trying to understand this theorem from Munkres:
If $G$ is the free product $G_1*G_2$, and $N_1$,$N_2$ are some normal subgroups of $G_1$ and $G_2$, respectively, and $N$ is the least normal subgroup of $G$ containing $N_1$ and $N_2$, then
$G/N \cong (G_1/N_1)*(G_2/N_2)$
I cannot convince myself that $G-> (G_1*G2)/N$ will map $N_1$ to identity. Can anyone see why that holds ?