Kerodon

Example 11.3.0.13. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be a unitary lax functor, and let $U: \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ denote the forgetful functor of Notation 5.6.1.11. If $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $Y$ is an object of the category $\mathscr {F}(D)$, let $\widetilde{f}_{Y}$ denote the pair $(f, \operatorname{id}_{ \mathscr {F}(f)(Y) } )$, which we regard as a morphism from $( C, \mathscr {F}(f)(Y) )$ to $(D, Y)$ in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$. Then the construction $(f,Y) \mapsto \widetilde{f}_{Y}$ is a cleavage of $U$ (see Example 11.10.3.7), which we will refer to as the tautological cleavage.