$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 11.9.4.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between locally small $\infty $-categories, and let $\gamma : \mathscr {H}_{\operatorname{\mathcal{C}}}(-,-) \rightarrow \mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), F(-) )$ be as in Proposition 11.9.4.1. Then:
The natural transformation $\gamma $ exhibits the functor $\mathscr {H}_{\operatorname{\mathcal{D}}}( F(-), - ): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ as a left Kan extension of $\mathscr {H}_{\operatorname{\mathcal{C}}}$ along the functor $(\operatorname{id}\times F): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$.
The natural transformation $\gamma $ exhibits the functor $\mathscr {H}_{\operatorname{\mathcal{D}}}(-, F(-) ): \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as a left Kan extension of $\mathscr {H}_{\operatorname{\mathcal{C}}}$ along the functor $(F^{\operatorname{op}} \times \operatorname{id}): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.
Proof.
We will prove the second assertion; the second follows by a similar argument. By virtue of the commutative diagram (11.17) and the transitivity of left Kan extensions (Proposition 7.3.8.18), it will suffice to prove the following:
- $(1)$
The natural transformation $\alpha $ exhibits $\mathscr {H}_{\operatorname{\mathcal{C}}}(-,-)$ as a left Kan extension of $\underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ along the forgetful functor $\operatorname{Tw}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$.
- $(2)$
The natural transformation $\beta |_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ exhibits the functor $\mathscr {H}_{\operatorname{\mathcal{D}}}( -, F(-) )$ as a left Kan extension of $\underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ along the composite functor
\[ \operatorname{Tw}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\xrightarrow { F^{\operatorname{op}} \times \operatorname{id}} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. \]
Assertion $(1)$ is a special case of Proposition 8.3.5.6. Assertion $(2)$ follows from Proposition 8.3.4.20, since the natural transformation $\beta |_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ exhibits the profunctor $\mathscr {H}_{\operatorname{\mathcal{D}}}( -, F(-) )$ as represented by $F$ (see the proof of Proposition 8.3.4.15).
$\square$