Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.2.3.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

$(1)$

If $F$ is a left adjoint, then it is left cofinal.

$(2)$

If $F$ is a right adjoint, then it is right cofinal.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Suppose that $F$ admits a right $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. For every object $X \in \operatorname{\mathcal{D}}$, Corollary 6.2.4.2 guarantees that the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X}$ has a final object. In particular, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X}$ is weakly contractible (Corollary 4.6.6.26). Allowing $X$ to vary and applying Theorem 7.2.3.1, we conclude that $F$ is left cofinal. $\square$