Example 7.1.7.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{E}}$. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $U$-cocartesian (see Definition 5.1.1.1).
- $(2)$
The morphism $f$ is a $U$-colimit diagram when viewed as a map of simplicial sets $(\Delta ^0)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.
- $(3)$
The morphism $f$ is $U_{X/}$-initial when viewed as an object of the slice $\infty $-category $\operatorname{\mathcal{E}}_{X/}$, where $U_{X/}: \operatorname{\mathcal{E}}_{X/} \rightarrow \operatorname{\mathcal{C}}_{U(X)/ }$ is the functor induced by $U$.
- $(4)$
The morphism $f$ is $V$-initial when viewed as an object of the oriented fiber product $\{ X\} \vec{\times }_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}$, where $V: \{ X\} \vec{\times }_{ \operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \{ U(X) \} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ 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 4.6.4.20.