Proposition 8.3.5.6. 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 8.3.5.1.
- $(2)$
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 $\{ \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \} \operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$