Definition 1.2.8 (Inverse Limit). Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an downward-directed system, then an inverse limit of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$, denoted $\lim_{\longleftarrow}A_{i}$, is a pair $(A, \bracsn{f^A_i}_{i \in I})$ such that:

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

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

   \[\xymatrix{ A_i \ar@{->}[r]^{f^i_j} & A_j \\ A \ar@{->}[u]^{f^A_i} \ar@{->}[ru]_{f^A_j} & }\]

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

   \[\xymatrix{ & A_i \\ B \ar@{->}[r]_{S} \ar@{->}[ru]^{g^B_i} & A \ar@{->}[u]_{f^A_i} }\]

   for all $i \in I$.