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Remark 5.5.2.9 (Fibers of the Forgetful Functor). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories. For every object $C \in \operatorname{\mathcal{C}}$, there is a canonical isomorphism of categories

\[ \mathscr {F}(C) \simeq \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}, \]

which carries each object $X \in \mathscr {F}(C)$ to the object $(C,X) \in \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and each morphism $u: X \rightarrow Y$ in $\mathscr {F}$ to the morphism $(\operatorname{id}_{C}, u \circ \epsilon _{C}(X)): (C,X) \rightarrow (C,Y)$ of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (here $\epsilon _{C}: \mathscr {F}(\operatorname{id}_ C) \simeq \operatorname{id}_{ \mathscr {F}(C)}$ denotes the identity constraint on the functor $\mathscr {F}$). Similarly, for each functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$, we have a canonical isomorphism

\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{\mathcal{C}}} \int ^{\operatorname{\mathcal{C}}} \mathscr {F}. \]