$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.7.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then:
- $(1)$
If $X$ is final as an object of the $\infty $-category $\operatorname{\mathcal{C}}$, then it is also final when viewed as an object of the homotopy category $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$.
- $(2)$
If $\operatorname{\mathcal{C}}$ has an final object and $X$ is final as an object of the homotopy category $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$, then $X$ is final as an object of the $\infty $-category $\operatorname{\mathcal{C}}$.
Proof.
Assertion $(1)$ is immediate from the definition. To prove $(2)$, assume that $\operatorname{\mathcal{C}}$ has a final object $Y$. Then $Y$ is also final when viewed as an object of the homotopy category $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$. If $X$ is a final object of $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$, then $X$ and $Y$ are isomorphic when viewed as objects of $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$, hence also when viewed as objects of the $\infty $-category $\operatorname{\mathcal{C}}$. Invoking Corollary 4.7.3.16, we conclude that $X$ is also a final object of $\operatorname{\mathcal{C}}$.
$\square$