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Remark 5.5.1.6. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a functor. Then the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ fits into a pullback diagram

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

Here $\operatorname{Set}_{\ast }$ denotes the category of pointed sets (see Example 4.2.3.3).