Kerodon

Proof. Let $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by the representable functors. By virtue of Theorem 8.2.5.4, the construction $Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, Y)$ determines an equivalence of $\infty $-categories $h^{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. It follows that postcomposition with $h^{\bullet }$ induces an equivalence of $\infty $-categories

which is a restatement of Proposition 8.2.6.1. $\square$

Proof. We have a commutative diagram of $\infty $-categories

where the vertical maps are cocartesian fibrations and the functor $T$ carries $\lambda _{+}$-cocartesian morphisms of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ to $\pi $-cocartesian morphisms of the $\infty $-category $\{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}})$ (see Corollary 8.1.1.12). By virtue of Corollary 7.2.3.15, the functor $T$ is left cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the induced map

is left cofinal; here the oriented fiber product on the right is formed with respect to the functor $\mathscr {K}(-, X): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$. Proposition 8.1.2.1 guarantees that the identity morphism $\operatorname{id}_{X}$ is initial when viewed as an object of the $\infty $-category $\infty $-category $\operatorname{Tw}( \operatorname{\mathcal{D}}) \times _{ \operatorname{\mathcal{D}}} \{ X\} $: that is, so that the inclusion map $\{ \operatorname{id}_ X \} \hookrightarrow \operatorname{Tw}( \operatorname{\mathcal{D}}) \times _{ \operatorname{\mathcal{D}}} \{ X\} $ is left cofinal (Example 7.2.1.4). Using Proposition 7.2.1.6, we see that $T_{X}$ is left cofinal if and only if the inclusion map $\{ \beta ( \operatorname{id}_ X ) \} \hookrightarrow \{ \Delta ^{0} \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{C}}^{\operatorname{op}}$ is left cofinal: that is, if and only if $\beta ( \operatorname{id}_ X )$ is initial when viewed as an object of the $\infty $-category $\{ \Delta ^{0} \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{C}}^{\operatorname{op}}$. By virtue of Proposition 5.6.6.22, this is equivalent to the requirement that $\beta ( \operatorname{id}_{X} )$ exhibits the functor $\mathscr {K}(-,X)$ as represented by the object $G(X) \in \operatorname{\mathcal{C}}$. $\square$

Proof. Let $\operatorname{\mathcal{E}}$ denote the oriented fiber product $\{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}})$ and let $\mu : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$ be the projection onto the second factor, so that we have a tautological natural transformation $\widetilde{\beta }: \underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {K} \circ \mu $. It follows from Proposition 7.6.2.15 that $\widetilde{\beta }$ exhibits $\mathscr {K}$ as a left Kan extension of $\underline{\Delta ^0}_{\operatorname{\mathcal{E}}}$ along $\mu $. The natural transformation $\beta $ then determines a functor $T: \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{E}}$ such that precomposition with $T$ carries $\widetilde{\beta }$ to $\beta $. By the transitivity of the formation of of Kan extensions (Proposition 7.3.7.18), we are reduced to showing that the identity transformation $\operatorname{id}: \underline{ \Delta ^0}_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) } \rightarrow \underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}} \circ T$ exhibits $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$ as a left Kan extension of $\underline{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ along $T$. This is a special case of Remark 7.6.2.12, since the functor $T$ is left cofinal (Proposition 8.2.6.12). $\square$

Proof. We first show that $(1)$ implies $(2)$. Since $\operatorname{\mathcal{C}}$ is locally small, it admits a $\operatorname{Hom}$-functor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ (Proposition 8.2.3.10). Choose a natural transformation $\alpha : \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{C}})} \rightarrow \mathscr {H}$ which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. Then $\alpha $ exhibits the profunctor $\mathscr {H}$ as represented by the identity functor $\operatorname{id}_{ \operatorname{\mathcal{C}}}$ (Example 8.2.6.9). By virtue of Remark 8.2.6.11, we may assume that the profunctor $\mathscr {K}$ is given by the composition

Applying Remark 8.2.6.14, we see that precomposition with the functor $\operatorname{Tw}(G): \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{Tw}( \operatorname{\mathcal{C}})$ carries $\alpha $ to a natural transformation

which exhibits $\mathscr {K}$ as represented by $G$.

The implication $(2) \Rightarrow (3)$ follows from Proposition 8.2.6.13. It follows that $(1)$ implies $(3)$, and the reverse implication follows from the uniqueness of Kan extensions up to isomorphism (Remark 7.3.6.6). $\square$