Kerodon

Example 5.6.1.7 (Set-Valued Functors). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of sets. Then we can also regard $\mathscr {F}$ as a functor from $\operatorname{\mathcal{C}}$ to the $2$-category $\mathbf{Cat}$ (by composing with the fully faithful embedding $\operatorname{Set}\hookrightarrow \mathbf{Cat}$, carrying each set $S$ to the associated discrete category). In this case, the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ of Definition 5.6.1.1 agrees with the category of elements of $\mathscr {F}$ defined in Construction 5.2.6.1. Similarly, for every functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$, the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ can be identified with the category of elements of $\mathscr {F}$ defined in Variant 5.2.6.2.