Definition 1.2.7 (Direct Limit). Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system, then a direct limit of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$, denoted $\lim_{\longrightarrow}A_{i}$, is a pair $(A, \bracsn{f^i_A}_{i \in I})$ such that:

1. For each $i \in I$, $f^{i}_{A} \in \mor{A_i, A}$.

2. For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:

   \[\xymatrix{ A_i \ar@{->}[rd]_{f^i_A} \ar@{->}[r]^{f^i_j} & A_j \ar@{->}[d]^{f^j_A} \\ & A }\]

3. For any pair $(B, \bracsn{g^i_B}_{i \in I})$ satisfying (1) and (2), there exists a unique $g \in \mor{A, B}$ such that the following diagram commutes

   \[\xymatrix{ A_j \ar@{->}[d]_{f^j_A} \ar@{->}[rd]^{g^i_B} & \\ A \ar@{->}[r]_{g} & B }\]

   for all $i \in I$.