Corollary 5.4.7.12. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $Z$ be an object of $\operatorname{\mathcal{C}}$. Then:
- $(1)$
The projection map $\pi : \operatorname{\mathcal{C}}_{/Z} \rightarrow \operatorname{\mathcal{C}}$ induces a cartesian fibration of $\infty $-categories $\operatorname{Pith}(\pi ): \operatorname{Pith}(\operatorname{\mathcal{C}}_{/Z} ) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$.
- $(2)$
A morphism $u$ of $\operatorname{Pith}(\operatorname{\mathcal{C}}_{/Z})$ is $\operatorname{Pith}(\pi )$-cartesian if and only if it corresponds to a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (in this case, it is also $\pi $-cartesian when viewed as a morphism of $\operatorname{\mathcal{C}}_{/Z}$).
- $(3)$
The inclusion $\operatorname{Pith}(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism from $\operatorname{Pith}(\operatorname{\mathcal{C}})_{/Z}$ to the (non-full) subcategory of $\operatorname{Pith}(\operatorname{\mathcal{C}}_{/Z} )$ spanned by the $\pi $-cartesian morphisms.