Kerodon

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

Proposition 6.2.1.17. 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 the unit of an adjunction. Then, 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 $\mathrm{h} \mathit{\operatorname{Kan}}$; here the second map is given by the composition law of Construction 4.6.3.9.

Proof. It will suffice to show that, for every Kan complex $T$, the induced amp

\begin{eqnarray*} \pi _0( \operatorname{Fun}(T, \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) )) & = & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }(T, \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) ) \\ & \xrightarrow {\theta } & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Kan}}}(T, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) )) \\ & = & \pi _0( \operatorname{Fun}(T, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) ) ) ) \end{eqnarray*}

is bijective. Let $\underline{C} \in \operatorname{Fun}(T, \operatorname{\mathcal{C}})$ and $\underline{D} \in \operatorname{Fun}(T, \operatorname{\mathcal{D}})$ be the constant morphisms taking the values $C$ and $D$, respectively. Unwinding the definitions, we see that $\theta $ can be identified with the map

\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Fun}(T,\operatorname{\mathcal{D}})}}( F \circ \underline{C}, \underline{D} ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(T,\operatorname{\mathcal{C}})}}( \underline{C}, G \circ \underline{D} ) \]

given by the formation of right adjuncts with respect to the homotopy class $[\eta ]$ (regarded as a $2$-morphism in the category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$). The bijectivity of $\theta $ now follows from the criterion of Proposition 6.1.2.9. $\square$