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Proposition 8.2.3.12. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category, let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor, and let $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ be a natural transformation. The following conditions are equivalent:

$(1)$

The natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$: that is, it satisfies condition $(\ast )$ of Definition 8.2.3.2.

$(2)$

The diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [dr]_-{\mathscr {H}} \ar@ {<=}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [ur]^{\lambda } \ar [rr]_{ \underline{ \Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} } & & \operatorname{\mathcal{S}}. } \]

exhibits $\mathscr {H}$ as a left Kan extension of the constant functor $\underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) }$ along the left fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.

$(3)$

The pair $( \mathscr {H}, \alpha )$ is initial when viewed as an object of the oriented fiber product $\{ \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \} \operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 7.6.2.15 and Remark 8.2.3.8. Since $\operatorname{\mathcal{C}}$ is locally small, Proposition 8.2.3.10 guarantees that the functor $\underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) }$ admits a left Kan extension along $\lambda $, so the equivalence $(2) \Leftrightarrow (3)$ follows from Corollary 7.3.6.5. $\square$