Definition 8.3.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ denote the left fibration of Proposition 8.1.1.11, and let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. We say that a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if it satisfies the following condition:
- $(\ast )$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the natural transformation $\alpha $ induces a homotopy equivalence of Kan complexes
\[ \alpha _{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {H}(X,Y) ). \]