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Example 7.1.4.10. 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 7.1.3.8 (and also of Example 7.1.1.5, by virtue of Remark 7.1.4.6).