Let $\{C_i\}_{i \in I}$ be a set of objects in a category $\mathcal{C}$ A direct sum of the collection $\{C_i\}_{i \in I}$ is an object $\coprod_{i \in I} C_i$ of $\mathcal{C}$ with morphisms $\iota_i: C_i \longrightarrow \coprod_{j \in I} C_j$ for each $i \in I$ such that:

For every object $A$ in $\mathcal{C}$ and any collection of morphisms $f_i: C_i \longrightarrow A$ for every $i \in I$ there exists a unique morphism $f: \coprod_{i \in I} C_i \longrightarrow A$ making the following diagram commute for all $i \in I$ $$ \xymatrix{ C_i \ar[dr]_{\iota_i} \ar[rr]^{f_i} & & A \\ & \coprod_{j \in I} C_j \ar@{-->}[ur]_{f} } $$