Corollary 5.2.7.5. Let $\operatorname{\mathcal{C}}$ be a category. Then:
Construction 5.2.6.1 determines a fully faithful functor
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}, \]whose essential image consists of the left covering functors $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.
Variant 5.2.6.2 determines a fully faithful functor
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int ^{\operatorname{\mathcal{C}}} \mathscr {F}, \]whose essential image consists of the right covering functors $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.