Definition 23.2 (Coproduct of abelian groups or the direct sum):

Given a family of abelian groups $\{G_{\alpha}\;/\;\alpha \in \Lambda\}$, their coproduct or direct sum is an abelian group $G$ together with a family of group homomorphisms $\{\iota_{\alpha}:G_{\alpha}\longrightarrow G\;/\;\alpha \in \Lambda\}$ such that the following universal property holds.

Given any abelian group $A$ and a family of group homomorphisms $f_{\alpha}:G_{\alpha}\longrightarrow A$, there exists a unique group homomorphism $\phi : G\longrightarrow A$ such that each of the diagrams commutes:

$$\xymatrix{ G_{\alpha} \ar[rr]^{\iota_{\alpha}}\ar[rd]_{f_{\alpha}} & & G \ar[ld]^{\phi}\\ & A }$$