Kerodon

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

Remark 8.2.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Using Remark 8.2.3.8, we see that a functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if and only if it is a covariant transport representation for the left fibration $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$. In other words, $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if $\lambda $ can be lifted to an equivalence of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ with the $\infty $-category of elements $\int _{ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}} \mathscr {H}$.