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Remark 5.6.1.13. Let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. If $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ is a functor of $2$-categories, then the composition $(\mathscr {F} \circ V ): \operatorname{\mathcal{D}}\rightarrow \mathbf{Cat}$ is also a functor of $2$-categories, and we have a pullback diagram of categories

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{D}}}( \mathscr {F} \circ V ) \ar [r] \ar [d] & \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \ar [d] \\ \operatorname{\mathcal{D}}\ar [r]^-{V} & \operatorname{\mathcal{C}}} \]

where the vertical maps are the forgetful functors of Notation 5.6.1.11. Similarly, for every functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$, we have a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \int ^{\operatorname{\mathcal{D}}}( \mathscr {F} \circ V^{\operatorname{op}} ) \ar [r] \ar [d] & \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \ar [d] \\ \operatorname{\mathcal{D}}\ar [r]^-{V} & \operatorname{\mathcal{C}}. } \]