Kerodon

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

Corollary 6.2.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between $\infty $-categories, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. The following conditions are equivalent:

$(1)$

The natural transformation $\eta $ is the unit of an adjunction between $F$ and $G$.

$(2)$

For every pair of objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (G \circ F)(C), G(D) ) \xrightarrow { \circ [ \eta _ C]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(D) ) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

$(3)$

The functor $F$ admits a right adjoint. Moreover, for every pair of objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{D}}}}( F(C), D) \xrightarrow {G} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( (G \circ F)(C), G(D) ) \xrightarrow { \circ [ \eta _ C]} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, G(D) ) \]

is a bijection.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 6.2.1.17. Suppose that condition $(2)$ is satisfied. Then, for every object $D \in \operatorname{\mathcal{D}}$, the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D)$ is representable by the object $G(D) \in \operatorname{\mathcal{D}}$. Applying Proposition 6.2.4.5, we deduce that $F$ admits a right adjoint, so that condition $(3)$ is satisfied. We now complete the proof by showing that $(3) \Rightarrow (1)$. Note that, if condition $(3)$ is satisfied, then the natural transformation $\eta $ exhibits $\mathrm{h} \mathit{G}: \mathrm{h} \mathit{\operatorname{\mathcal{D}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as a right adjoint of the functor $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ (see Variant 6.1.2.11). Invoking Proposition 6.2.1.14, we deduce that $\eta $ is the unit of an adjunction between $F$ and $G$. $\square$