Definition 11.10.3.1. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{D}}$ having image $\overline{f}: \overline{X} \rightarrow \overline{Y}$ in the category $\operatorname{\mathcal{C}}$.
We say that $f$ is locally $U$-cartesian if, for every object $W$ of the fiber category $\operatorname{\mathcal{D}}_{ \overline{X} } = \{ \overline{X} \} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, postcomposition with $f$ induces a bijection
\[ \operatorname{Hom}_{ \operatorname{\mathcal{D}}_{\overline{X}} }( W, X) \xrightarrow {f \circ } \{ f' \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Y): U(f') = U(f) \} . \]We say that $f$ is locally $U$-cocartesian if, for every object $Z$ of the fiber category $\operatorname{\mathcal{D}}_{ \overline{Y} } = \{ \overline{Y} \} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, precomposition with $f$ induces a bijection
\[ \operatorname{Hom}_{ \operatorname{\mathcal{D}}_{\overline{Y}} }( Y, Z) \xrightarrow {\circ f} \{ f' \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Z): U(f') = U(f) \} . \]