In James R. Munkres's "Topology" he has the lemma
"Lemma 68.5. Let $\{G_{\alpha}\}_{\alpha \in J}$ be a family of groups; let $G$ be a group; let $i_{\alpha}: G_{\alpha} \to G$ be a family of homomorphisms. If the extension condition of Lemma $68.3$ holds, then each $i_{\alpha}$ is a monomorphism and $G$ is the free product of the groups $i_{\alpha}(G_{\alpha})$."
"Lemma 68.3. Let $\{G_{\alpha}\}_{\alpha \in J}$ be a family of groups; let $G$ be a group; let $i_{\alpha}: G_{\alpha} \to G$ be a family of homomorphisms. If each $i_{\alpha}$ is a monomorphism and $G$ is the free product of the groups $i_{\alpha}(G_{\alpha})$, then $G$ satisfies the following condition:
Given a group $H$ and family of homomorphisms $h_{\alpha}: G_{\alpha} \to H$, there exists a homomorphism $h: G\to H$ such that $h \circ i_{\alpha}=h_{\alpha}$ for each $\alpha$.
Furthermore h is unique."
The proof of Lemma 68.5. uses the following lemma
"Lemma 68.4. Let $\{G_{\alpha}\}_{\alpha \in J}$ be a family of groups. Suppose $G$ and $G'$ are groups and $i_{\alpha}: G_{\alpha} \to G$ and $i_{\alpha}': G_{\alpha} \to G'$ are families of monomorphisms such that the families $\{i_{\alpha}(G_{\alpha})\}$ and $\{i_{\alpha}(G_{\alpha}')\}$ generate $G$ and $G'$, respectively. If both $G$ and $G'$ have the extension property stated in the preceding lemma, then there is a unique isomorphism $\phi: G\to G'$ such that $\phi \circ i_{\alpha} = i_{\alpha}'$ for all $\alpha$."
My question is whether the statement of Lemma 68.5. is missing the additional assumption that the family $\{i_{\alpha}(G_{\alpha})\}$ generates $G$? Or is that implicitly implied by the condition in Lemma 68.3.?
