Homomorphisms in the definition of free products In the book "a Course in the Theory of Groups" of Derek J.S. Robinson, the free product of groups $ \{G_{\lambda} \vert\ \lambda \in \Lambda \} $ is a group $ G $ and a collection of homomorphisms $ \{ \iota_{\lambda} : G_{\lambda}  \vert\ \lambda \in \Lambda \}  $  such that given any group $ H $ and a homomorphisms $ \{ \phi_{\lambda} : G_{\lambda} \to H \} $, we have a unique homomorphism $ \phi : G \to H $ such that $ \iota_{\lambda} \phi  = \phi_{\lambda}$.
My question is that are these maps $ \iota_{\lambda} $ the standard injections, i.e. that for all $ \lambda \in \Lambda $,  $ \iota_{\lambda} (g) = g $ for all $g \in G_{\lambda} $?
 A: What you describe is the universal property of a coproduct. There is no explicit construction in your text, only the universal property, and depending on the construction the coproduct maps will be defined differently.
Here is one construction which works. You form a big set $String(\bigsqcup_\lambda UG_\lambda)$ of strings with symbols from the set $\bigsqcup_\lambda UG_\lambda$. You turn that set of strings into monoid by concatination. Next, you define functions $i_\lambda: G_\lambda\to String(\bigsqcup_\lambda UG_\lambda)$ which send a $g\in G_\lambda$ to the string $g$ of length one. Next, you make the least amount of identifications on the set of strings which are necessary so that all the $i_\lambda$s become group homomorphism. The resulting set $String(\bigsqcup_\lambda UG_\lambda)/\sim$ with its induced monoid structure is in fact a group, and together with the maps $G_\lambda \to String(\bigsqcup_\lambda UG_\lambda) \to String(\bigsqcup_\lambda UG_\lambda)/\sim$ it is the coproduct and satisfies the universal property described in your question. Now you have a concrete description of the group morphisms $i_\lambda$.
