Example 4.3.1.2. Let $\operatorname{Set}$ denote the category of sets, and let $S \in \operatorname{Set}$ be a set. Then the construction
\[ (f: X \rightarrow S) \mapsto \{ X_{s} = f^{-1} \{ s\} \} _{s \in S} \]
induces an equivalence of categories $\operatorname{Set}_{/S} \rightarrow \prod _{s \in S} \operatorname{Set}$.