Definition 1.2.2 (Product). Let $\catc$ be a category and $\seqi{A}\subset \obj{\catc}$. A product of $\seqi{A}$ is a pair $(P, \seqi{\pi})$ where

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

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

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

   \[\xymatrix{ C \ar@{->}[rd]^{f_i} \ar@{->}[d]_{f} & \\ P \ar@{->}[r]_{\pi_i} & A_i }\]

   for all $i \in I$.