Definition 8.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that a profunctor
\[ \mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}} \]
is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if it is a covariant transport representation for the twisted arrow coupling $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.