Example 11.10.2.2. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be a unitary lax functor, let $U: \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ denote the forgetful functor of Notation 5.6.1.11, and equip $U$ with the tautological cleavage described in Example 11.3.0.13. Then, for every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the contravariant transport functor $f^{\ast }$ of Proposition 5.2.2.1 fits into a (strictly) commutative diagram of categories
\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}(D) \ar [r]^-{ \mathscr {F}(f) } \ar [d]^-{\sim } & \mathscr {F}(C) \ar [d]^-{\sim } \\ (\int ^{\operatorname{\mathcal{C}}} \mathscr {F})_{D} \ar [r]^-{ f^{\ast } } & (\int ^{\operatorname{\mathcal{C}}} \mathscr {F})_{C}, } \]
where the vertical maps are the isomorphisms of Remark 5.6.1.12.