Remark 5.2.6.7 (01QE)

Remark 5.2.6.7. Let $\operatorname{\mathcal{C}}$ be a small category, let $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ be the category of set-valued functors on $\operatorname{\mathcal{C}}^{\operatorname{op}}$, and let $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ be the Yoneda embedding (so that $h$ carries each object $C \in \operatorname{\mathcal{C}}$ to the representable functor $h_{C} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , C)$). For any object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$, the category of elements $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ fits into a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \ar [d] \ar [r] & \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})_{/\mathscr {F}} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{h} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}). } \]

This is essentially a reformulation of Yoneda's lemma (see Corollary 8.4.2.7 for an $\infty $-categorical counterpart).