8.3.4 Representable Profunctors
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. There is a fully faithful embedding from the category of functors $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ to the category of profunctors $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{Set})$, which assigns to each functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ the representable profunctor
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) ). \]
This construction has an $\infty $-categorical counterpart:
Proposition 8.3.4.1 (Classification of Representable Profunctors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Let $\kappa $ be an uncountable cardinal for which $\operatorname{\mathcal{C}}$ is locally $\kappa $-small, and let
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, -): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \]
be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ (see Notation 8.3.3.7). Then the construction $G \mapsto \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( -, G(-) )$ determines a fully faithful functor
\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}^{<\kappa } ), \]
whose essential image is spanned by the representable profunctors from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$.
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 8.3.3.13, 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
\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})), \]
which is a restatement of Proposition 8.3.4.1.
$\square$
Definition 8.3.4.2. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small and let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(-, -): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. We say that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is representable by $G$ if it isomorphic to the composition
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow {\operatorname{id}\times G} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(-, -) } \operatorname{\mathcal{S}}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) ) \]
as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}^{<\kappa })$. By virtue of Proposition 8.3.3.2, this condition does not depend on the choice of $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$.
Example 8.3.4.3. Let $\operatorname{\mathcal{C}}$ be a locally $\kappa $-small $\infty $-category, and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ be a functor. Then $\mathscr {F}$ is representable by an object $X \in \operatorname{\mathcal{C}}$ (in the sense of Variant 5.6.6.2) if and only if, when regarded as a profunctor from $\Delta ^0$ to $\operatorname{\mathcal{C}}$, it is representable by the functor $\Delta ^0 \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 8.3.4.2).
Interchanging the roles of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, we obtain the following dual notion:
Variant 8.3.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{D}}$ is locally $\kappa $-small and let $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(-, -): \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{D}}$. We say that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $F$ if it isomorphic to the composition
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow {F^{\operatorname{op}} \times \operatorname{id}} \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{D}}}(-, -) } \operatorname{\mathcal{S}}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), Y) \]
as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}^{<\kappa })$. By virtue of Proposition 8.3.3.2, this condition does not depend on the choice of $\operatorname{Hom}$-functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, - )$.
Example 8.3.4.5. Let $\operatorname{\mathcal{C}}$ be a locally $\kappa $-small $\infty $-category and let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ be a profunctor from $\operatorname{\mathcal{C}}$ to itself. The following conditions are equivalent:
The profunctor $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.
The profunctor $\mathscr {H}$ is representable by the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (Definition 8.3.4.2).
The profunctor $\mathscr {H}$ is corepresentable by the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (Variant 8.3.4.4).
Example 8.3.4.8. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be locally $\kappa $-small $\infty $-categories, and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$. The following conditions are equivalent:
The profunctor $\mathscr {K}$ is balanced (Definition 8.3.2.18).
The profunctor $\mathscr {K}$ is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ which is an equivalence of $\infty $-categories.
The profunctor $\mathscr {K}$ is corepresentable by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ which is an equivalence of $\infty $-categories.
By virtue of Theorem 8.3.3.13, this is a reformulation of Corollary 8.3.2.20.
Proposition 8.3.4.9 (Adjunctions as Profunctors). Let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor of $\infty $-categories which represents a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{<\kappa }$. Then a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ is left adjoint to $G$ if and only if it corepresents the profunctor $\mathscr {K}$. In particular, $\mathscr {K}$ is corepresentable if and only if the functor $G$ admits a left adjoint.
Proof.
Choose a realization of $\mathscr {K}$ as the covariant transport representation of a coupling of $\infty $-categories $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ (see Remark 8.3.2.5). By virtue of Remark 8.3.4.6, the coupling $\lambda $ is representable by the functor $G$. By virtue of Theorem 8.2.5.1, a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ is left adjoint to $G$ if and only if it corepresents the coupling $\lambda $. Invoking Remark 8.3.4.6 again, we see that this is equivalent to the requirement that $F$ corepresents the profunctor $\mathscr {K}$.
$\square$
Recall that, if $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $\infty $-categories, then we write $\operatorname{LFun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{+} )$ for the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}_{-},\operatorname{\mathcal{C}}_{+} )$ spanned by those functors $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ which are left adjoints (Notation 6.2.1.3). Similarly, we write $\operatorname{RFun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )$ for the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )$ spanned by those functors $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ which are right adjoints.
Corollary 8.3.4.10. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. Let $\kappa $ be an uncountable cardinal for which $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are locally $\kappa $-small, and let $\operatorname{\mathcal{E}}\subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}^{< \kappa } )$ denote the full subcategory spanned by those profunctors $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ which are both representable and corepresentable. Then:
- $(1)$
Composition with the covariant Yoneda embedding $\operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{ < \kappa } )$ induces an equivalence of $\infty $-categories $\rho : \operatorname{RFun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} ) \rightarrow \operatorname{\mathcal{E}}$.
- $(2)$
Composition with the contravariant Yoneda embedding $\operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}^{< \kappa } )$ induces an equivalence of $\infty $-categories $\lambda : \operatorname{LFun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{+} ) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$.
- $(3)$
The composition $[\rho ]^{-1} \circ [\lambda ^{\operatorname{op}}]$ determines a canonical isomorphism $\operatorname{LFun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{+} )^{\operatorname{op}} \simeq \operatorname{RFun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )$ in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$, which carries each functor $F \in \operatorname{LFun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{+} )$ to a functor $G \in \operatorname{RFun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )$ which is right adjoint to $F$.
Proof.
Assertion $(1)$ follows by combining Propositions 8.3.4.1 and 8.3.4.9, and assertion $(2)$ follows by a similar argument. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with Proposition 8.3.4.9.
$\square$
For many applications, Definition 8.3.4.2 is insufficiently precise. Given a functor of $\infty $-categories $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, we would like to be able to consider not only profunctors which are representable by $G$ (meaning that they are abstractly isomorphic to the profunctor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, G(Y) )$) but profunctors which are represented by $G$ (meaning that we have chosen an isomorphism with the profunctor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,G(Y) )$, or some essentially equivalent datum). Here it is inconvenient that the functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(-,-)$ is well-defined only up to isomorphism. To address this point, it is convenient to encode representability in a different way.
Notation 8.3.4.11. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1). We will regard the contractible Kan complex $\Delta ^0$ as an object of $\operatorname{\mathcal{S}}$. For every $\infty $-category $\operatorname{\mathcal{E}}$, we let $\underline{ \Delta ^0}_{\operatorname{\mathcal{E}}}$ denote the constant functor $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$ taking the value $\Delta ^0$.
Definition 8.3.4.12. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$, and let $\mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$ denote the composite functor
\[ \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { G^{\operatorname{op}} \times \operatorname{id}} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { \mathscr {K} } \operatorname{\mathcal{S}}. \]
Suppose we are given a natural transformation $\beta : \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) } \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$, where $\underline{\Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$ denotes the constant functor $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}}$ taking the value $\Delta ^0$. We say that $\beta $ exhibits the profunctor $\mathscr {K}$ as represented by $G$ if, for every object $D \in \operatorname{\mathcal{D}}$, the evaluation of $\beta $ at the object $\operatorname{id}_{D} \in \operatorname{Tw}( \operatorname{\mathcal{D}})$ determines a vertex $\beta ( \operatorname{id}_{D} ) \in \mathscr {K}( G(D), D )$ which exhibits the functor $\mathscr {K}( -, D)$ as represented by the object $G(D) \in \operatorname{\mathcal{C}}$ (see Variant 5.6.6.2).
Example 8.3.4.14. In the situation of Definition 8.3.4.12, suppose that $\operatorname{\mathcal{D}}= \Delta ^0$. In this case, we can identify the profunctor $\mathscr {K}$ with a functor $K: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$, we can identify the functor $G$ with an object $X \in \operatorname{\mathcal{C}}$, and we can identify $\beta $ with a vertex of the Kan complex $K(X)$. Then $\beta $ exhibits the profunctor $\mathscr {K}$ as represented by the functor $G$ (in the sense of Definition 8.3.4.12) if and only if it exhibits the functor $K$ as represented by the object $X$ (in the sense of Variant 5.6.6.2.
Proposition 8.3.4.15. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is locally small, and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. The following conditions are equivalent:
- $(1)$
The profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is representable by $G$, in the sense of Definition 8.3.4.2.
- $(2)$
There exists a natural transformation $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ which exhibits $\mathscr {K}$ as represented by $G$, in the sense of Definition 8.3.4.12.
Proof.
By virtue of Remarks 8.3.4.6 and 8.3.4.13, this follows by applying Proposition 8.2.4.3 to the coupling
\[ \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. \]
$\square$
Variant 8.3.4.16 (Corepresentable Profunctors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. We say that a natural transformation $\beta : \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{C}}) } \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{C}})}$ exhibits $\mathscr {K}$ as corepresented by $F$ if, for every object $X \in \operatorname{\mathcal{C}}$, the image $\beta ( \operatorname{id}_{X} ) \in \mathscr {K}( X, F(X) )$ exhibits the functor $\mathscr {K}( X, - ): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by the object $F(X) \in \operatorname{\mathcal{D}}$, in the sense of Definition 5.6.6.1. Equivalently, $\beta $ exhibits $\mathscr {K}$ as corepresented by $F$ if it exhibits $\mathscr {K}$ as represented by the opposite functor $F^{\operatorname{op}}$, when regarded as a profunctor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to $\operatorname{\mathcal{D}}^{\operatorname{op}}$ (see Remark 8.3.2.3).
See Remark 5.6.6.4.
Proposition 8.3.4.19. Suppose we are given a functor of $\infty $-categories $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, and a natural transformation $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$. Then $\beta $ exhibits $\mathscr {K}$ as represented by $G$ (in the sense of Definition 8.3.4.12) if and only if the induced map $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}})$ is left cofinal.
Proof.
By virtue of Remark 8.3.4.13, this is a special case of Proposition 8.2.4.9.
$\square$
Proposition 8.3.4.20 (Representable Profunctors as Kan Extensions). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor, and let $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ be a natural transformation which exhibits $\mathscr {K}$ as represented by $G$. Then $\beta $ exhibits $\mathscr {K}$ as a left Kan extension of the constant diagram $\underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ along the composite map
\[ \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { G^{\operatorname{op}} \times \operatorname{id}} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. \]
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 7.6.2.17 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 7.3.8.18), 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 7.6.2.13, since the functor $T$ is left cofinal (Proposition 8.3.4.19).
$\square$
Corollary 8.3.4.21 (The Universal Mapping Property of Representable Profunctors). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Suppose we are given a pair of profunctors $\mathscr {K}, \mathscr {K}': \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, and let $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ be a natural transformation which exhibits $\mathscr {K}$ as represented by $G$. Then precomposition with $\beta $ induces a homotopy equivalence of Kan complexes
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {K}, \mathscr {K}') \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{Tw}( \operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})}( \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}})}, \mathscr {K}'|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} ). \]
Proof.
Combine Propositions 8.3.4.20 and 7.3.6.1.
$\square$
Example 8.3.4.22 (Spaces of Natural Transformation). Let $G,G': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\mathscr {H}$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. Combining Corollary 8.3.4.21 with Proposition 8.3.4.1, we obtain homotopy equivalences of Kan complexes
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})}( G, G' ) & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {H} \circ (\operatorname{id}\times G), \mathscr {H} \circ (\operatorname{id}\times G') ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})}( \underline{\Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}, \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}}) }) \\ & \simeq & \varprojlim ( \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}}) }). \end{eqnarray*}
Stated more informally, the space of natural transformations from $G$ to $G'$ can be viewed as a limit of the diagram
\[ \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad (f: X \rightarrow Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(X), G'(Y)). \]