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Definition 8.2.3.2 ($\operatorname{Hom}$-Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\lambda : \operatorname{Tw}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ be the left fibration of Proposition 8.1.1.11. A $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ is a pair $( \mathscr {H}, \alpha )$, where $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a functor and $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ is a natural transformation which 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

\[ \alpha _{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {H}(X,Y) ). \]