7.74 Let \( r:\left\{A_{i}, \alpha_{j}^{i}\right\} \rightarrow\left\{B_{i}, \beta_{j}^{i}\right\} \) and \( s:\left\{B_{i}, \beta_{j}^{i}\right\} \rightarrow\left\{C_{i}, \gamma_{j}^{i}\right\} \) be transformations of inverse systems over an index set \( I \). If
\[
0 \rightarrow A_{i} \xrightarrow{r_{i}} B_{i} \xrightarrow{s_{i}} C_{i}
\]
is exact for each \( i \in I \), prove that there is an exact sequence
\[
0 \rightarrow \lim _{\longleftarrow} A_{i} \xrightarrow{\vec{r}} \underset{\leftrightarrows}{\lim } B_{i} \xrightarrow{\vec{s}} \lim _{\longleftarrow} C_{i} .
\]
