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Remark 5.5.2.5. 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 construction $(C \in \operatorname{\mathcal{C}}) \mapsto \mathscr {F}(C)^{\operatorname{op}}$ determines a functor of $2$-categories $\mathscr {F}^{\operatorname{op}}: \operatorname{\mathcal{C}}= (\operatorname{\mathcal{C}}^{\operatorname{op}})^{\operatorname{op}} \rightarrow \mathbf{Cat}$. In this case, we have a canonical isomorphism of categories

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

where the left hand side is given by Definition 5.5.2.3 and the right hand side is given by Definition 5.5.2.1.