Corollary 8.1.2.21. 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:
- $(1)$
The morphism $f$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.
- $(2)$
The morphism $f$ is initial when regarded as an object of the $\infty $-category $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$.
- $(3)$
The morphism $f$ is initial when regarded as an object of the $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} $.