Usually, for a family $\{G_\alpha\}_{\alpha \in A}$ of abelian groups, one defines $ \bigoplus_{\alpha \in A} G_\alpha \:= \{ a \in \prod_{\alpha \in A} G_\alpha : \sharp \{ \alpha \in A : \pi_\alpha(a) \ne 0 \} < \infty \} < \prod_{\alpha \in A} G_\alpha $ where $\pi_\beta : \prod_{\alpha \in A} G_\alpha \to G_\beta$ is the projection. One has the inclusion maps $i_\beta : G_\beta \to \bigoplus_{\alpha \in A} G_\alpha$ such that $\pi_\alpha \circ i_\alpha = \mbox{id}$ and $\pi_\alpha \circ i_\beta = 0$ for $\alpha \ne \beta$. It can then be shown that $\bigoplus_{\alpha \in A} G_\alpha$ has the following universal property:
(a) For any abelian group $H$ and any family of homomorphisms $\phi_\beta : G_\beta \to H$ there is a unique homomorphism $\phi : \bigoplus_{\alpha \in A} G_\alpha \to H$ such that $\phi \circ i_\alpha = \phi_\alpha$.
From the definition of $\bigoplus_{\alpha \in A} G_\alpha$, it follows that
(b) Any element $g \in \bigoplus_{\alpha \in A} G_\alpha$ can be written as $i_{i_1}(g_{\alpha_1})+...+i_n(g_{\alpha_n})$, where $\{\alpha_1,...,\alpha_n\} \subset A$ is finite and $g_{\alpha_1} \in G_{\alpha_1}$,...,$g_{\alpha_n} \in G_{\alpha_n}$.
If we define a direct sum of the $G_\alpha$s as a couple $(G,\{j_\alpha\}_{\alpha \in A})$ with $G$ and $j_\alpha : G_\alpha \to G$ with the property (a), can we show that (b) is true for $G$ and the $j_\alpha$s, without using an explicit construction of the group $G$, and without appealing to the existence of an isomorphism between $G$ and $\bigoplus_{\alpha \in A} G_\alpha$?