Definition 1.2.3 (Coproduct). Let $\catc$ be a category and $\seqi{A}\subset \obj{\catc}$. A product of $\seqi{A}$ is a pair $(P, \seqi{\iota})$ where

1. $P \in \obj{\catc}$.

2. For each $i \in I$, $\iota_{i} \in \mor{A_i P}$.

3. For any pair $(C, \seqi{f})$ satisfying (1) and (2), there exists a unique $f \in \mor{P, C}$ such that the following diagram commutes

   \[\xymatrix{ & C \\ A_i \ar@{->}[r]_{\iota_i} \ar@{->}[ru]^{f_i} & P \ar@{->}[u]_{f} }\]

   for all $i \in I$.