Definition 1.2.6 (Directed System). Let $\catc$ be a category and $(I, \lesssim)$ be a directed set. A directed system is a pair $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ such that:

1. $\seqi{A}\subset \obj{\catc}$.

2. For each $i \in I$, $f^{i}_{i} = \text{Id}_{A_i}$.

3. For each $i, j \in I$ with $i \lesssim j$, $f^{i}_{j} \in \mor{A_i, A_j}$.

4. For each $i, j, k \in I$ with $i \lesssim j \lesssim k$, $f^{j}_{k} \circ f^{i}_{j} = f^{i}_{k}$.

If $I$ is upward/downward-directed, then $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is upward/downward-directed.