Example 7.1.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The morphism $f$ is an isomorphism.
When regarded as a morphism $(\Delta ^0)^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, $f$ is a limit diagram.
When regarded as a morphism $(\Delta ^0)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, $f$ is a colimit diagram.
This is a restatement of Proposition 4.6.7.22 (and also of Example 7.1.1.5, by virtue of Remark 7.1.3.6).