Remark 5.2.6.9 (01QP)

Remark 5.2.6.9. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor, and let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the category of elements of $\mathscr {F}$ (Construction 5.2.6.1). Then the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering functor, in the sense of Definition 4.2.3.1. This follows from the pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d] & \operatorname{Set}_{\ast } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{Set}} \]

of Remark 5.2.6.6, together with Remark 4.2.3.6 and Example 4.2.3.3. We will see in §5.2.7 that the converse is also true: for every left covering functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ and isomorphism $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is compatible with the functor $U$ (Corollary 5.2.7.5). By virtue of Proposition 5.2.6.8, the functor $\mathscr {F}$ is unique up to canonical isomorphism.