$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 6.2.4.1. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Then $G$ admits a left adjoint if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the following condition is satisfied:
- $(\ast _{X})$
There exists an object $Y \in \operatorname{\mathcal{D}}$ and a morphism $u: X \rightarrow G(Y)$ in $\operatorname{\mathcal{C}}$ such that, for every object $Z \in \operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(Y,Z) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(Y), G(Z) ) \xrightarrow { \circ [u]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Z) ) \]
is a homotopy equivalence of Kan complexes.
Proof.
We first prove necessity. Suppose that there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which exhibits $F$ as a left adjoint of $G$. Fix an object $X \in \operatorname{\mathcal{C}}$ and set $Y = F(X)$. Then $\eta $ determines a morphism $\eta _{X}: X \rightarrow G(Y)$ which satisfies the requirement of condition $(\ast _ X)$ (Proposition 6.2.1.17).
We now prove sufficiency. Let $\operatorname{\mathcal{E}}$ denote the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ be the cartesian fibration of Proposition 5.2.3.15. Let us abuse notation by identifying the fibers $\{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ with $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. Fix an object $X \in \operatorname{\mathcal{C}}$, and suppose that there exists an object $Y \in \operatorname{\mathcal{D}}$ together with a morphism $u: X \rightarrow G(Y)$ satisfying the requirement of condition $(\ast _ X)$. Then we can identify $u$ with a morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{E}}$. Our assumption on $u$ guarantees that the morphism $f$ is $U$-cocartesian (see Corollary 5.1.2.3). Consequently, if condition $(\ast _ X)$ is satisfied for every object $X \in \operatorname{\mathcal{C}}$, then $U$ is a cocartesian fibration. Applying Proposition 6.2.3.4, we conclude that $G$ admits a left adjoint.
$\square$