Kerodon

Remark 11.4.0.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often abuse terminology by referring to a functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if there exists a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{\operatorname{Tw}(\operatorname{\mathcal{C}})}$ which satisfies condition $(\ast )$ of Definition 8.3.5.1. In this case, we will say that $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.