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

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


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


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 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 Allowing $X$ to vary and applying Theorem, we conclude that $F$ is left cofinal. $\square$