Theorem 68.7 Let $ G = G_1 * G_2 $. Let $ N_i $ be a normal subgroup of $ G_i $, for $ i = 1, 2 $. If $ N $ is the least normal subgroup of $ G $ that contains $ N_1 $ and $ N_2 $, then $$ G/N \simeq \left(G_1/N_1\right) * \left(G_2/N_2\right). $$
Proof. The composite of the inclusion and projection homomorphisms $$ G_1 \longrightarrow G_1 * G_2 \longrightarrow (G_1 * G_2)/N $$
carries $N_1$ to the identity element, so that it induces a homomorphism $$ i_1 : G_1/N_1 \longrightarrow (G_1 * G_2)/N. $$
Similarly, the composite of the inclusion and projection homomorphisms induces a homomorphism $$ i_2 : G_2/N_2 \longrightarrow (G_1 * G_2)/N. $$ We show that the extension condition of Lemma 68.5 holds with respect to $i_1$ and $i_2$; it follows that $i_1$ and $i_2$ are monomorphisms and that $(G_1 * G_2)/N$ is the external free product of $G_1/N_1$ and $G_2/N_2$ relative to these monomorphisms.
So let $ h_1 : G_1/N_1 \to H \ \text{and} \ h_2 : G_2/N_2 \to H $ be arbitrary homomorphisms. The extension condition for $G_1*G_2$ implies that there is a homomorphism of $G_{1}*G_{2}$ into $H$ that equals the composite
$$ G_i \to G_i / N_i \to H $$
of the projection map and $h_i$ on $G_i$, for $i = 1, 2$. This homomorphism carries the elements of $N_{1}$ and $N_{2}$ to the identity element, so its kernel contains $N$. Therefore it induces a homomorphism $ h : (G_{1} * G_{2})/N \to H $ that satisfies the conditions $ h_{1} = h \circ i_{1} \ \text{and} \ h_{2} = h \circ i_{2}. $
the extension condition which it is referring to is :
Lemma 68.1. Let $ G $ be a group; let $\{G_\alpha\}$ be a family of subgroups of $ G $. If $ G $ is the free product of the groups $G_\alpha$, then $ G $ satisfies the following condition:
- Given any group $ H $ and any family of homomorphisms $ h_\alpha : G_\alpha \rightarrow H $, there exists a homomorphism $ h : G \rightarrow H $ whose restriction to $ G_\alpha $ equals $ h_\alpha $, for each $\alpha$.
Furthermore, $ h $ is unique.
My problem about it is that for the extension condition for $G_1\ast G_2$ we should have $h_i : G_i \to H$ not $h_i : G_i/N_i \to H$? can someone please help me get it through. am I missing something?