Proposition 1.2.10. Let $R$ be a ring and $(\seqi{A}, \bracs{T^i_j|i, j \in I, i \lesssim j)}$ be a downward-directed system of $R$-modules, then there exists $(A, \bracsn{T^A_i}_{i \in I})$ such that:
For each $i \in I$, $T^{A}_{i} \in \hom(A; A_{i})$.
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ A_i \ar@{->}[r]^{T^i_j} & A_j \\ A \ar@{->}[u]^{T^A_i} \ar@{->}[ru]_{T^A_j} & }\]For any pair $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom(B; A)$ such that the following diagram commutes
\[\xymatrix{ & A_i \\ B \ar@{->}[r]_{S} \ar@{->}[ru]^{S^B_i} & A \ar@{->}[u]_{T^A_i} }\]for all $i \in I$.
Proof. Let
For each $i \in I$, let $T^{A}_{i} = \pi_{i}$, then $(A, \bracsn{T^A_i}_{i \in I})$ satisfies (1) and (2) by definition of $A$.
(U): Let $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2). Let
then for any $x \in B$ and $i, j \in I$ with $i \lesssim j$,
so $S \in \hom(B; A)$, and the diagram commutes. Since any map $f: B \to A$ is uniquely determined by its composition with the projections, $S$ is unique.$\square$