Example 7.1.6.9. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $U$-cartesian (see Definition 5.1.1.1).
- $(2)$
The morphism $f$ is a $U$-limit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.
- $(3)$
The morphism $f$ is $U_{/Y}$-final when viewed as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$, where $U_{/Y}: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / U(Y) }$ is the functor induced by $U$.
- $(4)$
The morphism $f$ is $V$-final when viewed as an object of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $, where $V: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ U(Y) \} $ is the functor induced by $U$.
The equivalence $(2) \Leftrightarrow (3)$ is immediate from the definition, the equivalence $(1) \Leftrightarrow (3)$ follows from Remark 7.1.6.8 and Proposition 5.1.1.14, and the equivalence $(3) \Leftrightarrow (4)$ follows from Corollary 4.6.4.18.