Kerodon

Notation 5.6.1.11. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories. Then the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is equipped with a forgetful functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$, given on objects by the construction $(C,X) \mapsto C$ and on morphisms by the construction $(f,u) \mapsto f$. Similarly, for every functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$, the category of $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ of Definition 5.6.1.4 is equipped with a forgetful functor $U: \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.