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Corollary 5.3.7.11. 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 $\operatorname{Pith}(q)$-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.

Proof. Assertions $(1)$ and $(2)$ follow from Corollary 5.3.7.10, and assertion $(3)$ is an immediate consequence of $(2)$. $\square$