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Corollary 5.3.7.10. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let

\[ q': \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}\quad \quad q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}} \]

be the projection maps. Then:

$(1)$

The functor $\operatorname{Pith}(q): \operatorname{Pith}( \operatorname{\mathcal{C}}_{/f} ) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ is a cartesian fibration of $\infty $-categories. Moreover, a morphism $u$ of $\operatorname{Pith}(\operatorname{\mathcal{C}}_{/f})$ is $\operatorname{Pith}(q)$-cartesian if and only if, for every vertex $z \in K$, the composite map

\[ \Delta ^2 \simeq \Delta ^1 \star \{ z\} \hookrightarrow \Delta ^1 \star K \xrightarrow {u} \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(2)$

The functor $\operatorname{Pith}(q'): \operatorname{Pith}( \operatorname{\mathcal{C}}_{f/} ) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ is a cocartesian fibration of $\infty $-categories. Moreover, a morphism $v$ of $\operatorname{Pith}(\operatorname{\mathcal{C}}_{f/})$ is $\operatorname{Pith}(q')$-cocartesian if and only if, for every vertex $x \in K$, the composite map

\[ \Delta ^2 \simeq \{ x\} \star \Delta ^1 \hookrightarrow K \star \Delta ^1 \xrightarrow {v} \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.