Kerodon

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

Example 6.2.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. For every object $D \in \operatorname{\mathcal{D}}$, the construction $(C \in \operatorname{\mathcal{C}}) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D)$ determines an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor from $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}}$ to the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. If the functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, then this functor is representable by the object $G(D) \in \operatorname{\mathcal{C}}$ (see Proposition 6.2.1.17).