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8.2.4 Presentations of Representable Couplings

For some applications, it is convenient to work with a variant of Definition 8.2.3.5.

Definition 8.2.4.1. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor. We will say that a morphism of couplings

8.26
\begin{equation} \begin{gathered}\label{equation:representable-coupling-witness-revised} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [r]^-{ \widetilde{G} } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{+} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} } \end{gathered} \end{equation}

exhibits $\lambda $ as represented by $G$ if, for every object $C \in \operatorname{\mathcal{C}}_{+}$, the image $\widetilde{G}( \operatorname{id}_{C} )$ is a universal object of $\operatorname{\mathcal{C}}$.

Remark 8.2.4.2. In the situation of Definition 8.2.4.1, if the diagram (8.26) exhibits $\lambda $ as represented by $G$ if and only if the functor $\widetilde{G}$ carries each isomorphism in $\operatorname{\mathcal{C}}_{+}$ (regarded as an object of the $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{C}}_{+})$) to a universal object of $\operatorname{\mathcal{C}}$. This follows from Remark 8.2.1.2, since every isomorphism in $\operatorname{\mathcal{C}}_{+}$ is isomorphic to an identity morphism (when viewed as an object of $\operatorname{Tw}(\operatorname{\mathcal{C}}_{+})$).

Proposition 8.2.4.3. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor. The following conditions are equivalent:

$(1)$

The coupling $\lambda $ is representable by $G$ (in the sense of Definition 8.2.3.1).

$(2)$

There exists a morphism of couplings

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [r]^-{ \widetilde{G} } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{+} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} } \]

which exhibits $\lambda $ as represented by $G$ (in the sense of Definition 8.2.4.1).

Proof. We first show that $(2)$ implies $(1)$. Suppose that there exists a morphism of couplings

8.27
\begin{equation} \begin{gathered}\label{equation:duality-functor-left-to-right-fix} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [r]^-{ \widetilde{G} } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{+} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} } \end{gathered} \end{equation}

which exhibits $\lambda $ as represented by $G$ (in the sense of Definition 8.2.4.1). For each object $C_{+} \in \operatorname{\mathcal{C}}_{+}$, the functor $\widetilde{G}$ carries $\operatorname{id}_{C}$ to a universal object of $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{+}(C) = C_{+}$. It follows that the coupling $\lambda $ is representable. Theorem 8.2.3.4, guarantees that there exists a functor $G': \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ such that $\lambda $ is representable by $G'$. Choose morphism of couplings

8.28
\begin{equation} \begin{gathered}\label{equation:duality-functor-left-to-right2-fix} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{G}' } \ar [d]^{\lambda } & \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ \operatorname{id}\times G' } & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{-}, } \end{gathered} \end{equation}

which exhibits $\lambda $ as represented by $G'$ (in the sense of Definition 8.2.3.5). Composing (8.28) with (8.27), we obtain a morphism of twisted arrow couplings

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}_{+} ) \ar [r]^-{ \widetilde{G}' \circ \widetilde{G} } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \ar [d] \\ \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times G'} & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} } \]

where the functor $\widetilde{G}' \circ \widetilde{G}$ carries isomorphisms of $\operatorname{\mathcal{C}}_{+}$ to isomorphisms of $\operatorname{\mathcal{C}}_{-}$. Invoking Corollary 8.2.2.11, we deduce that the functors $G$ and $G'$ are isomorphic, so that $\lambda $ is also representable by $G$ (Remark 8.2.3.2).

We now show that $(1)$ implies $(2)$. Assume that $\lambda $ is representable by $G$. Setting $G' = G$, we can choose a diagram (8.28) which exhibits $\lambda $ as represented by $G$. Applying Proposition 8.2.3.9, we deduce that composition with $\widetilde{G}'$ induces a homotopy equivalence of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \{ G\} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+} ), \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{+} ) } \{ \operatorname{id}\} \ar [d]^{\theta } \\ \{ G\} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+} ), \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} ) } \{ G \} . } \]

In particular, there exists an object $(G, \widetilde{G}, \operatorname{id}) \in \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+} ), \operatorname{\mathcal{C}})$ such that $(\operatorname{id}, \widetilde{G}', G) \circ (G, \widetilde{G}, \operatorname{id})$ is isomorphic to $(G, \operatorname{Tw}(G), G)$ in the $\infty $-category $\operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}), \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ) )$. In particular, the functor $\widetilde{G}' \circ \widetilde{G}: \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}_{-})$ is isomorphic to the functor $\operatorname{Tw}(G)$, and therefore carries isomorphisms of $\operatorname{\mathcal{C}}_{+}$ to isomorphisms of $\operatorname{\mathcal{C}}_{-}$. It follows that the functor $\widetilde{G}: \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \rightarrow \operatorname{\mathcal{C}}$ carries isomorphisms in $\operatorname{\mathcal{C}}_{+}$ to universal objects of $\operatorname{\mathcal{C}}$, so that the diagram (8.27) exhibits $\lambda $ as represented by $G$. $\square$

Corollary 8.2.4.4. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ and $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ be couplings of $\infty $-categories which are representable by functors $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ and $H: \operatorname{\mathcal{D}}_{+} \rightarrow \operatorname{\mathcal{D}}_{-}$, respectively. Let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{D}}_{-}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{D}}_{+}$ be functors. The following conditions are equivalent:

$(1)$

The functors $H \circ F_{+}$ and $F_{-} \circ G$ are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{-} )$. That is, the diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F_{+} } \ar [d]^{ G } & \operatorname{\mathcal{D}}_{+} \ar [d]^{H} \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{ F_{-} } & \operatorname{\mathcal{D}}_{-} } \]

commutes up to isomorphism.

$(2)$

There is a morphism of couplings

8.29
\begin{equation} \begin{gathered}\label{equation:commutativity-as-morphism-of-coupling0} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{F} } \ar [d]^{\lambda } & \operatorname{\mathcal{D}}\ar [d]^{\mu } \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}^{+} \ar [r]^-{ F_{-}^{\operatorname{op}} \times F_{+} } & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}, } \end{gathered} \end{equation}

where the functor $\widetilde{F}$ carries universal objects of $\operatorname{\mathcal{C}}$ to universal objects of $\operatorname{\mathcal{D}}$.

Proof. Choose morphisms of couplings

8.30
\begin{equation} \begin{gathered}\label{equation:commutativity-as-morphism-of-coupling} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \ar [r]^-{ \widetilde{G} } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{\lambda } & \operatorname{\mathcal{D}}\ar [r]^-{ \widetilde{H} } \ar [d]^{\mu } & \operatorname{Tw}( \operatorname{\mathcal{D}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} & \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} \ar [r]^-{ \operatorname{id}\times H } & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{-}, } \end{gathered} \end{equation}

which exhibit $\lambda $ and $\mu $ as represented by $G$ and $H$, respectively. We first prove that $(2)$ implies $(1)$. Suppose there exists a diagram (8.29), where $\widetilde{F}$ carries universal objects of $\operatorname{\mathcal{C}}$ to universal objects of $\operatorname{\mathcal{D}}$. Composing with the morphisms (8.30), we obtain a morphism of twisted arrow couplings

\[ \xymatrix@R =50pt@C=100pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \ar [r]^-{ \widetilde{H} \circ \widetilde{F} \circ \widetilde{G} } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ (F_{-} \circ G)^{\operatorname{op}} \times (H \circ F_{+}) } & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{-}, } \]

where the functor $\widetilde{H} \circ \widetilde{F} \circ \widetilde{G}$ carries isomorphisms in $\operatorname{\mathcal{C}}_{+}$ to isomorphisms in $\operatorname{\mathcal{D}}_{-}$. Applying Corollary 8.2.2.11, we deduce that the functors $F_{-} \circ G$ and $H \circ F_{+}$ are isomorphic.

We now show that $(1)$ implies $(2)$. Since $\lambda $ is representable and the twisted arrow pairing $\operatorname{Tw}(\operatorname{\mathcal{D}}_{-}) \rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{-}$ is corepresentable, Theorem 8.2.2.9 guarantees that there exists a morphism of pairings

8.31
\begin{equation} \begin{gathered}\label{equation:commutativity-as-morphism-of-coupling2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{\lambda } \ar [r]^-{ \widetilde{T} } & \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F^{\operatorname{op}}_{-} \times T_{+} } & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{-}, } \end{gathered} \end{equation}

where $\widetilde{T}$ carries universal objects of $\operatorname{\mathcal{C}}$ to isomorphisms in the $\infty $-category $\operatorname{\mathcal{D}}_{-}$. Composing with the pairing on the left half of (8.30), we obtain a morphism of twisted arrow pairings

\[ \xymatrix@R =50pt@C=75pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \ar [r]^-{ \widetilde{T} \circ \widetilde{G} } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ (F_{-} \circ G)^{\operatorname{op}} \times T_{+} } & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{-}. } \]

Applying Corollary 8.2.2.11, we conclude that $T_{+}$ is isomorphic to the functor $F_{-} \circ G$. If condition $(1)$ is satisfied, then $T_{+}$ is also isomorphic to the functor $H \circ F_{+}$. Replacing (8.31) by an isomorphic objects of $\infty $-category $\{ F_{-} \} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-})^{\operatorname{op}}} \times \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} ) )$, we may assume without loss of generality that $T_{+}$ is equal to $H \circ F_{+}$. Invoking the universal property of Proposition 8.2.3.9, we can further assume that (8.31) factors as a composition

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{\lambda } \ar [r]^-{ \widetilde{F} } & \operatorname{\mathcal{D}}\ar [r]^-{\widetilde{H} } \ar [d]^{\mu } & \operatorname{Tw}(\operatorname{\mathcal{D}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F^{\operatorname{op}}_{-} \times F_{+} } & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+} \ar [r]^-{\operatorname{id}\times H} & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{-}. } \]

Since $\widetilde{T} = \widetilde{H} \circ \widetilde{F}$ carries universal objects of $\operatorname{\mathcal{D}}$ to isomorphisms in $\operatorname{\mathcal{D}}_{-}$, the functor $\widetilde{F}$ carries universal objects of $\operatorname{\mathcal{C}}$ to universal objects of $\operatorname{\mathcal{D}}$. $\square$

Variant 8.2.4.5. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ be a functor. We will say that a morphism of couplings

8.32
\begin{equation} \begin{gathered}\label{equation:representable-coupling-witness-revised2} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \ar [r]^-{ \widetilde{F} } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{-} \ar [r]^-{\operatorname{id}\times F} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} } \end{gathered} \end{equation}

exhibits $\lambda $ as corepresented by $F$ if, for every object $X_{-} \in \operatorname{\mathcal{C}}_{-}$, the image $\widetilde{F}( \operatorname{id}_{X_{-}} )$ is a couniversal object of $\operatorname{\mathcal{C}}$. Equivalently, the diagram (8.32) exhibits $\lambda $ as corepresented by $F$ if exhibits the coupling $\lambda ': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{+} \times \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ of Remark 8.2.1.4 as represented by the functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}_{+}^{\operatorname{op}}$.

We now apply these ideas to prove a more precise version of Theorem 8.2.2.9. To (slightly) simplify the notation, we state the result in a dual form.

Theorem 8.2.4.6. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ and $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+}$ be couplings of $\infty $-categories. Assume that $\lambda $ is corepresentable by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ and that $\mu $ is representable by a functor $G: \operatorname{\mathcal{D}}_{+} \rightarrow \operatorname{\mathcal{D}}_{-}$. Then:

$(1)$

The coupling

\[ \Phi : \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \]

of Remark 8.2.2.3 is representable by the functor

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} ) \quad \quad T_{+} \mapsto G \circ T_{+} \circ F. \]
$(2)$

An object $( T_{-}, T, T_{+} ) \in \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is universal if and only if the functor $T$ carries couniversal objects of $\operatorname{\mathcal{C}}$ to universal objects of $\operatorname{\mathcal{D}}$.

Remark 8.2.4.7. Stated more informally, Theorem 8.2.4.6 states that if a coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is corepresented by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ and a coupling $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ is represented by a functor $G: \operatorname{\mathcal{D}}_{+} \rightarrow \operatorname{\mathcal{D}}_{-}$, then the objects of $\operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ can be identified with triples $(T_{-}, T_{+}, \alpha )$ where $T_{+}$ is a functor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{D}}_{+}$, and $\alpha : T_{-} \rightarrow G \circ T_{+} \circ F$ is a natural transformation of functors from $\operatorname{\mathcal{C}}_{-}$ to $\operatorname{\mathcal{D}}_{-}$. Moreover, the natural transformation $\alpha $ is an isomorphism if and only if the corresponding functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ carries couniversal objects to universal objects.

Example 8.2.4.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, let $(\lambda _{-}, \lambda _{+}): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ and $(\mu _{-},\mu _{+}): \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$ be the twisted arrow couplings of Example 8.2.0.2, and let

\[ \operatorname{ev}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}\quad \quad (F,C) \mapsto F(C) \]

be the evaluation functor. Passing to twisted arrow $\infty $-categories, we obtain a map

\[ \operatorname{Tw}(\operatorname{ev}): \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \times \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{D}}), \]

which we can identify with a functor $\widetilde{E}: \operatorname{Tw}( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$. By construction, the functor $\widetilde{E}$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \ar [d]^{\circ \lambda _{-}} & \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \ar [l] \ar [d]^{ \widetilde{T} } \ar [r] & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [d]^{\circ \lambda _{+}} \\ \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}^{\operatorname{op}} ) & \operatorname{Fun}( \operatorname{Tw}( \operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) ) \ar [l]_{\mu _{-} \circ } \ar [r]^-{\mu _{+} \circ } & \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}), } \]

and therefore determines a functor $E: \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$. The commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ) \ar [r]^-{E} \ar [d] & \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) ) \ar [d]^{\Phi } \\ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r]^-{ \operatorname{id}\times \operatorname{id}} & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) } \]

exhibits the coupling $\Phi $ as represented by the identity functor $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

Proof of Theorem 8.2.4.6. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling which is corepresented by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$, and let $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ be a coupling which is represented by a functor $G: \operatorname{\mathcal{D}}_{+} \rightarrow \operatorname{\mathcal{D}}_{-}$. It follows from Theorem 8.2.2.9 that the coupling

\[ \Phi : \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \]

of Remark 8.2.2.3 is representable, and that an object $(T_{-}, T, T_+) \in \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$ is universal if and only if the functor $T$ carries couniversal objects of $\operatorname{\mathcal{C}}$ to universal objects of $\operatorname{\mathcal{D}}$. We will complete the proof by showing that the coupling $\Phi $ is representable by the functor

\[ H: \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-}) \quad \quad T_{+} \mapsto G \circ T_{+} \circ F. \]

Choose a morphism of couplings

8.33
\begin{equation} \begin{gathered}\label{equation:internal-hom-coupling1} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{F} } \ar [d]^{\lambda } & \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} } \end{gathered} \end{equation}

which exhibits $\lambda $ as corepresented by $F$, and a morphism of couplings

8.34
\begin{equation} \begin{gathered}\label{equation:internal-hom-coupling2} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{D}}_{+}) \ar [r]^-{ \widetilde{G} } \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{\mu } \\ \operatorname{\mathcal{D}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+} \ar [r]^-{G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+} } \end{gathered} \end{equation}

which exhibits $\mu $ as represented by $G$.

Let $E: \operatorname{Tw}( \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) ) \rightarrow \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}), \operatorname{Tw}(\operatorname{\mathcal{D}}_{+} ) )$ be the comparison map of Example 8.2.4.8. Precomposition with (8.33) and postcomposition with (8.34) determines a functor $E': \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}), \operatorname{Tw}(\operatorname{\mathcal{D}}_{+} ) ) \rightarrow \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) ) \ar [r]^-{E' \circ E} \ar [d] & \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} )^{\operatorname{op}} \times \operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+}) \ar [r]^-{ H^{\operatorname{op}} \times \operatorname{id}} & \operatorname{Fun}(\operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times \operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+}) } \]

is commutative. We will complete the proof by showing that this diagram exhibits the coupling $\Phi $ as represented by $H$.

Fix a functor $T_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{D}}_{+}$; we wish to show that the the composite functor

\[ \operatorname{Tw}( \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) ) \xrightarrow {E} \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}), \operatorname{Tw}(\operatorname{\mathcal{D}}_{+} ) ) \xrightarrow {E'} \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]

carries $\operatorname{id}_{ T_{+} }$ to a universal object of $\operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Unwinding the definitions, we see that the image of $\operatorname{id}_{ T_{+} }$ is given by the triple $(G \circ T_{+} \circ F, \widetilde{G} \circ \operatorname{Tw}( T_{+} ) \circ \widetilde{F}, T_{+} ) \in \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Using the criterion of Theorem 8.2.2.9, we are reduced to showing that the composite functor

\[ \operatorname{\mathcal{C}}\xrightarrow { \widetilde{F} } \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \xrightarrow { \operatorname{Tw}( T_{+} ) } \operatorname{Tw}( \operatorname{\mathcal{D}}_{+} ) \xrightarrow { \widetilde{G} } \operatorname{\mathcal{D}} \]

carries every couniversal object $X \in \operatorname{\mathcal{C}}$ to a universal object of $\operatorname{\mathcal{D}}$. Proposition 8.2.3.7 guarantees that $\widetilde{F}(X) \in \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} )$ corresponds to an isomorphism in $\operatorname{\mathcal{C}}_{+}$, so its image under $\operatorname{Tw}( T_{+} )$ corresponds to an isomorphism in $\operatorname{\mathcal{D}}_{+}$; the desired result now follows from our hypothesis that the functor $\widetilde{G}$ carries isomorphisms in $\operatorname{\mathcal{D}}_{+}$ to universal objects of $\operatorname{\mathcal{D}}$. $\square$

We close this section by recording an alternative formulation of Definition 8.2.4.1:

Proposition 8.2.4.9. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories, let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor, and suppose we are given a morphism of couplings

8.35
\begin{equation} \begin{gathered}\label{equation:representability-as-left-cofinality} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [r]^-{ \widetilde{G} } \ar [d]^{\mu } & \operatorname{\mathcal{C}}\ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{+} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} } \end{gathered} \end{equation}

where $\mu = (\mu _{-}, \mu _{+})$ is the twisted arrow coupling of Example 8.2.0.2. The following conditions are equivalent:

$(1)$

The diagram (8.35) exhibits $\lambda $ as represented by $G$, in the sense of Definition 8.2.4.1. That is, the functor $\widetilde{G}$ carries isomorphisms of $\operatorname{\mathcal{C}}_{+}$ to universal objects of $\operatorname{\mathcal{C}}$.

$(2)$

The functor $\widetilde{G}$ is left cofinal.

Proof. By virtue of Proposition 8.2.1.7, the functors $\lambda _{+}$ and $\mu _{+}$ are cocartesian fibrations, and the functor $\widetilde{G}$ carries $\mu _{+}$-cocartesian morphisms of $\operatorname{Tw}(\operatorname{\mathcal{C}}_{+} )$ to $\lambda _{+}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$. By virtue of Corollary 7.2.3.16, the functor $\widetilde{G}$ is left cofinal if and only if, for every object $C \in \operatorname{\mathcal{C}}_{+}$, the induced map $\widetilde{G}_{C}: \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \times _{ \operatorname{\mathcal{C}}_{+} } \{ C \} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C\} $ is left cofinal. It will therefore suffice to show that the following conditions are equivalent, for each object $C \in \operatorname{\mathcal{C}}_{+}$.

$(1_ C)$

The image $\widetilde{G}( \operatorname{id}_ C )$ is a universal object of $\operatorname{\mathcal{C}}$: that is, it is initial when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C\} $.

$(2_ C)$

The functor $\widetilde{G}_{C}$ is left cofinal.

The equivalence $(1_ C) \Leftrightarrow (2_ C)$ is a special case of Corollary 7.2.1.9, since $\operatorname{id}_{C}$ is initial when viewed as an object of the $\infty $-category $\operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \times _{ \operatorname{\mathcal{C}}_{+} } \{ C \} $ (see Proposition 8.1.2.1). $\square$