Remark 4.3.1.3. Let $\operatorname{\mathcal{C}}$ be a category which admits finite limits and let $\ast $ denote a final object of $\operatorname{\mathcal{C}}$. For any object $S \in \operatorname{\mathcal{C}}$, one can adapt the construction of Example 4.3.1.2 to define a functor
\[ F: \operatorname{\mathcal{C}}_{/S} \rightarrow \prod _{s: \ast \rightarrow S} \operatorname{\mathcal{C}}\quad \quad F(X \rightarrow S) = \{ \ast \times _{S} X \} _{s: \ast \rightarrow S}. \]
Motivated by this observation, it is often useful to think of objects of the slice category $\operatorname{\mathcal{C}}_{/S}$ as “families” of objects of $\operatorname{\mathcal{C}}$ which are parametrized by $S$. Beware that the functor $F$ is usually not an equivalence of categories.