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Corollary 8.2.2.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The profunctor $\mathscr {K}$ is balanced (in the sense of Definition 8.2.2.11).

$(2)$

The $\infty $-category $\operatorname{\mathcal{D}}$ is locally small and $\mathscr {K}$ induces a fully faithful functor

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, - ), \]

whose essential image is spanned by the corepresentable functors $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$.

$(3)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small and $\mathscr {K}$ induces a fully faithful functor

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto \mathscr {K}(-, Y ), \]

whose essential image is spanned by the representable functors $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$.

Proof. We will prove the equivalence of $(1)$ and $(2)$; the equivalence of $(1)$ and $(3)$ follows by a similar argument. Assume first that $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is a balanced profunctor. Invoking Proposition 8.2.2.10, we see that the functor

\[ \Phi : \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, - ) \]

is fully faithful, and that the essential image of $\Phi $ consists of corepresentable functors from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{S}}$. Fix an object $Y \in \operatorname{\mathcal{D}}$. Since $\mathscr {K}$ is representable, there exists an object $X \in \operatorname{\mathcal{C}}$ and a vertex $\eta \in \mathscr {K}(X,Y)$ which exhibits the functor $\mathscr {K}( -, Y)$ as represented by $X$. Our assumption that $\mathscr {K}$ is balanced guarantees that $\eta $ also exhibits the functor $\mathscr {K}(X, -)$ as corepresented by $Y$. In particular, for every object $Y' \in \operatorname{\mathcal{D}}$, $\eta $ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(Y,Y') \xrightarrow {\sim } \mathscr {K}(X,Y')$, so that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y')$ is essentially small. If $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is any functor corepresented by $Y$, then Theorem 5.7.6.13 guarantees that $F$ is isomorphic to $\mathscr {K}(X, -)$ (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$), and therefore belongs to the essential image of $\Phi $. Allowing the object $Y$ to vary, we deduce that the profunctor $\mathscr {K}$ satisfies condition $(2)$.

We now prove the converse. Assume that the functor $\Phi $ is fully faithful and that the essential image of $\Phi $ is spanned by the corepresentable functors $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$. We wish to show that the profunctor $\mathscr {K}$ is balanced. Since $\Phi $ takes values in the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors, the profunctor $\mathscr {K}$ is corepresentable. We next show that $\mathscr {K}$ is representable. Fix an object $Y \in \operatorname{\mathcal{D}}$; we wish to show that the functor $\mathscr {K}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable. Since $\operatorname{\mathcal{C}}$ is locally small, there exists a functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by $Y$ (Theorem 5.7.6.13). Then $F$ belongs to the essential image of $\Phi $. We may therefore assume without loss of generality that $F = \mathscr {K}( X_0, -)$ for some object $X_0 \in \operatorname{\mathcal{C}}$. Choose a vertex $\eta _0 \in \mathscr {K}(X_0,Y) = F(Y)$ which exhibits the functor $F$ as corepresented by $Y$. Since $\Phi $ is fully faithful, Proposition 8.2.2.10 implies that $\eta _0$ also exhibits the functor $\mathscr {K}(-, Y)$ as represented by $X_0$.

To complete the proof, we must show that the pairing $\mathscr {K}$ satisfies the second condition of Definition 8.2.2.11. Let $Y \in \operatorname{\mathcal{D}}$ be as above, let $X$ be any object of $\operatorname{\mathcal{C}}$, and let $\eta $ be a vertex of the Kan complex $\mathscr {K}(X,Y)$. Assume that $\eta $ exhibits the functor $\mathscr {K}(-, Y)$ as represented by $X$; we wish to show that it also exhibits the functor $\mathscr {K}(X, -)$ as corepresented by $Y$ (the reverse implication follows from Proposition 8.2.2.10). Let $\eta _0 \in \mathscr {K}(X_0, Y)$ be as above. Since $\eta _0$ exhibits $\mathscr {K}(-, Y)$ as represented by $X_0$, there exists an isomorphism $u: X \rightarrow X_0$ in the $\infty $-category $\operatorname{\mathcal{C}}$ such that $\mathscr {K}(u,\operatorname{id}_ Y)( \eta _0 )$ and $\eta $ belong to the same connected component of the Kan complex $\mathscr {K}(X,Y)$ (Remark 5.7.6.6). We may therefore assume without loss of generality that $\eta = \mathscr {K}(u, \operatorname{id}_ Y)(\eta _0)$ (Remark 5.7.6.3). The desired result now follows by applying Remark 5.7.6.4 to the isomorphism of functors $\mathscr {K}(u, -): \mathscr {K}(X_0, -) \rightarrow \mathscr {K}(X, -)$. $\square$