Definition 11.3.0.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $f: C \rightarrow D$ be a morphism in the category $\operatorname{\mathcal{C}}$. A cleavage of $U$ over $f$ is a function
\[ \{ \textnormal{Objects of $\operatorname{\mathcal{E}}_{D}$} \} \rightarrow \{ \textnormal{Morphisms of $\operatorname{\mathcal{E}}$} \} \quad \quad Y \mapsto \widetilde{f}_{Y} \]
which associates to each object $Y \in \operatorname{\mathcal{E}}_{D}$ a locally $U$-cartesian morphism $\widetilde{f}_{Y}: X \rightarrow Y$ satisfying $U( \widetilde{f}_ Y) = f$.
A cleavage of $U$ consists of a choice, for each morphism $f$ of $\operatorname{\mathcal{C}}$, of a cleavage of $U$ over $f$. We will denote a cleavage of $U$ by $(f,Y) \mapsto \widetilde{f}_{Y}$.