Definition 23.1:

Given two groups $G_1$ and $G_2$, their coproduct is a group $G$ together with a pair of group homomorphisms $i_1:G_1\longrightarrow G$ and $i_2:G_2\longrightarrow G$ such that given any group $H$ and group homomorphisms $f_1:G_1\longrightarrow H$ and $f_2:G_2\longrightarrow H$ there exists a unique homomorphism $\phi:G\longrightarrow H$ such that

$$\phi\circ i_1 = f_1,\quad \phi\circ i_2 = f_2 \eqno(23.1)$$

summarized in the following diagram ($k = 1, 2.$):

$$\xymatrix{ G_k \ar[rr]^{i_k}\ar[rd]_{f_k} & & G \ar[ld]^{\phi}\\ & H }$$

The definition immediately generalizes to any arbitrary (not necessarily finite) collection of groups. The uniqueness clause in the definition is important and the following theorem hinges upon it.