# Kerodon

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### 8.2.6 Balanced Couplings

Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty$-categories. In this section, we formulate a concrete criterion to determine if $\lambda$ is representable (or corepresentable) by an equivalence of $\infty$-categories.

Definition 8.2.6.1. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty$-categories. We say that $\lambda$ is balanced if it satisfies the following conditions:

$(1)$

The coupling $\lambda$ is representable. That is, for each object $C_{+} \in \operatorname{\mathcal{C}}_{+}$, there exists a universal object $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{+}(C) = C_{+}$.

$(2)$

The coupling $\lambda$ is corepresentable. That is, for each object $C_{-} \in \operatorname{\mathcal{C}}_{-}$, there exists a couniversal object $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{-}(C) = C_{-}$.

$(3)$

An object $C \in \operatorname{\mathcal{C}}$ is universal if and only if it is couniversal.

Example 8.2.6.2. For every $\infty$-category $\operatorname{\mathcal{C}}$, the twisted arrow coupling $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ of Example 8.2.0.2 is balanced. See Example 8.2.1.5.

Example 8.2.6.3. Let $X$ be a Kan complex, let $\lambda _{-}: \operatorname{Fun}( \Delta ^1, X) \rightarrow X$ be the morphism given by evaluation at the vertex $0 \in \Delta ^1$, and let $\lambda _{+}: \operatorname{Fun}( \Delta ^1, X) \rightarrow X$ be the morphism given by evaluation at the vertex $1 \in \Delta ^1$. It follows from Corollary 3.1.3.3 that the map

$\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{Fun}( \Delta ^1, X) \rightarrow X \times X$

is a Kan fibration; in particular, we can view it as a coupling of $X$ with itself. For each vertex $x \in X$, the path spaces $\lambda _{-}^{-1} \{ x\} = \{ x\} \operatorname{\vec{\times }}_{X} X$ and $\lambda _{+}^{-1} \{ x\} = X \operatorname{\vec{\times }}_{X} \{ x\}$ are contractible Kan complexes (Example 3.4.1.13), so that every object of $\operatorname{Fun}( \Delta ^1, X)$ is both universal and couniversal for the coupling $\lambda$. In particular, $\lambda$ is a balanced coupling.

Remark 8.2.6.4. Suppose we are given a morphism of couplings

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{\lambda } & \operatorname{\mathcal{D}}\ar [d]^{\mu } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{F^{\operatorname{op}}_{-} \times F_{+}} & \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} }$

which is an equivalence (in the sense of Exercise 8.2.2.7). Then $\lambda$ is balanced if and only if $\mu$ is balanced. See Remark 8.2.2.8.

We can now formulate the main result of this section.

Theorem 8.2.6.5. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty$-categories. The following conditions are equivalent:

$(1)$

The coupling $\lambda$ is balanced.

$(2)$

The coupling $\lambda$ is representable by an equivalence of $\infty$-categories $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$.

$(3)$

The coupling $\lambda$ is corepresentable by an equivalence of $\infty$-categories $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$.

Corollary 8.2.6.6. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty$-categories. Then $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are equivalent if and only if there exists a balanced coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$.

Corollary 8.2.6.7. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty$-categories. Then $\lambda$ is balanced if and only if there exists an equivalence of couplings

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

Proof. Suppose that $\lambda$ is balanced. By virtue of Theorem 8.2.6.5, the coupling $\lambda$ is corepresentable by a functor $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ which is an equivalence of $\infty$-categories. We can therefore choose a categorical pullback square

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

which exhibits $\lambda$ as corepresented by $F_{-}$. Since $F_{-}$ is an equivalence of $\infty$-categories, it follows that $F$ is also an equivalence of $\infty$-categories (Proposition 4.5.2.21). The reverse implication is an immediate consequence of Example 8.2.6.2 (and Remark 8.2.6.4). $\square$

We will deduce Theorem 8.2.6.5 from the following more general result.

Proposition 8.2.6.8. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty$-categories which is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$. The following conditions are equivalent:

$(1)$

Every universal object of $\operatorname{\mathcal{C}}$ is couniversal.

$(2)$

The functor $G$ is fully faithful.

Proof. Using Proposition 8.2.4.3, we can choose a commutative diagram

8.52
$$\begin{gathered}\label{equation:coupling-fully-faithful} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \ar [r]^-{ \widetilde{G}'} \ar [d] & \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{G} } \ar [d]^{\lambda } & \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ \operatorname{id}\times G } & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{-}, } \end{gathered}$$

where the left square exhibits $\lambda$ as represented by $G$ in the sense of Definition 8.2.4.1, and the right square exhibits $\lambda$ as represented by $G$ in the sense of Definition 8.2.3.5. Invoking (the dual of) Lemma 8.2.3.6, we see that $(1)$ is equivalent to the following:

$(1')$

The left square of (8.52) is a categorical pullback diagram.

For each object $C \in \operatorname{\mathcal{C}}_{+}$, Proposition 8.2.3.7 guarantees that the composite functor $\widetilde{G} \circ \widetilde{G}'$ carries $\operatorname{id}_{C}$ to an isomorphism of $\operatorname{\mathcal{C}}_{-}$ (regarded as an object of $\operatorname{Tw}(\operatorname{\mathcal{C}}_{-} )$). It follows from Theorem 8.2.2.11 that $(G, \widetilde{G}' \circ \widetilde{G}, G)$ and $(G, \operatorname{Tw}(G), G)$ are isomorphic when viewed as objects of $\operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}), \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}))$ (since both are initial objects of the $\infty$-category $\operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}), \operatorname{Tw}(\operatorname{\mathcal{C}}_{-})) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-})} \{ G\}$). In particular, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}_{+}$, the diagram of Kan complexes

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

commutes up to homotopy, where the horizontal maps are the homotopy equivalences of Notation 8.1.2.14. Using Corollary 5.1.7.15, we see that $(2)$ is equivalent to the following:

$(2')$

The outer rectangle of (8.52) is a categorical pullback diagram.

The equivalence of $(1')$ and $(2')$ is a special case of Proposition 4.5.2.18, since the right square of (8.52) is a categorical pullback square by assumption. $\square$

Proof of Theorem 8.2.6.5. We will prove the equivalence $(1) \Leftrightarrow (2)$; the equivalence $(1) \Leftrightarrow (3)$ follows by a similar argument. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is a coupling of $\infty$-categories which is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$. Combining Theorem 8.2.5.1 with Proposition 8.2.6.8, we see that $\lambda$ is balanced if and only if the following conditions are satisfied:

• The functor $G$ is fully faithful.

• The functor $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$.

• The functor $F$ is fully faithful.

It follows from Corollary 6.2.2.19 that these conditions are satisfied if and only if $G$ is an equivalence of $\infty$-categories. $\square$

We close this section by describing an example of a balanced coupling which will play an important role in §8.6.

Proposition 8.2.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{ev}_{0}, \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the evaluation functors. Let $L$ be the collection of all morphisms $u$ in $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ such that $\operatorname{ev}_0(u)$ is an isomorphism in $\operatorname{\mathcal{C}}$, and let $R$ be the collection of all morphisms $u$ in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ such that $\operatorname{ev}_{1}(u)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then the maps $\operatorname{Cospan}( \operatorname{ev}_0 )$ and $\operatorname{Cospan}( \operatorname{ev}_1 )$ determine a balanced coupling

$\lambda : \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}}) \times \operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}}).$

The proof of Proposition 8.2.6.9 will require some preliminaries. The first step is to establish the following:

Lemma 8.2.6.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the morphism

$\lambda : \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}}) \times \operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}})$

is a left fibration of $\infty$-categories.

The proof of Lemma 8.2.6.10 is straightforward but somewhat tedious; we therefore defer the argument to §8.6.6, where we prove a more a general statement (Lemma 8.6.5.14). It follows from Lemma 8.2.6.10 that we can view the map

$\lambda : \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}}) \times \operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}})$

as a coupling of the $\infty$-category $\operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}})^{\operatorname{op}}$ with itself (see Remark 8.1.6.2). To deduce Proposition 8.2.6.9, we will compare $\lambda$ with the twisted arrow coupling $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of Example 8.2.0.2.

Construction 8.2.6.11. Let $Q$ be a partially ordered set and let $Q^{\operatorname{op}}$ denote the opposite partially ordered set. To avoid confusion, for each element $q \in Q$, we write $q^{\operatorname{op}}$ for the corresponding element of $Q^{\operatorname{op}}$. Let $\operatorname{Tw}(Q)$ denote the twisted arrow category of $Q$ (Example 8.1.0.5), which we identify with the partially ordered subset of $Q^{\operatorname{op}} \times Q$ consisting of those pairs $(p^{\operatorname{op}}, q)$ satisfying $p \leq q$. We then have a morphism of partially ordered sets $\xi _{Q}: \operatorname{Tw}(Q) \times [1] \rightarrow Q^{\operatorname{op}} \star Q$, given concretely by the formulae

$\xi _{Q}( p^{\operatorname{op}}, q, i ) = \begin{cases} p^{\operatorname{op}} & \text{ if } i = 0 \\ q & \text{ if } i = 1. \end{cases}$

Let $\operatorname{\mathcal{C}}$ be a simplicial set. For every nonempty finite linearly ordered set $Q$, we obtain a map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q), \operatorname{Tw}(\operatorname{\mathcal{C}}) ) & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q^{\operatorname{op}} \star Q), \operatorname{\mathcal{C}}) \\ & \xrightarrow { \circ \xi _{Q}} & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{Tw}(Q) \times [1]), \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}( \operatorname{N}_{\bullet }(Q) ) \times \Delta ^1, \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}( \operatorname{N}_{\bullet }(Q) ), \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q), \operatorname{Cospan}( \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) ) ). \end{eqnarray*}

This construction depends functorially on $Q$, and therefore determines a morphism of simplicial sets $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) )$.

Remark 8.2.6.12. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the morphism $\Xi$ of Construction 8.2.6.11 can be described concretely on low-dimensional simplices as follows:

• On vertices, $\Xi$ is given by the formula $\Xi (f) = f$. Here we abuse notation by identifying vertices of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ and $\operatorname{Cospan}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) )$ with edges of the simplicial set $\operatorname{\mathcal{C}}$.

• Let $e: f_0 \rightarrow f_1$ be an edge of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$, which we identify with a $3$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$ displayed informally in the diagram

$\xymatrix@C =50pt@R=50pt{ X_0 \ar [d]^{ f_0 } & X_1 \ar [l]_{g} \ar [d]^{f_1} \\ Y_0 \ar [r]^-{h} & Y_1. }$

Then $\Xi (e)$ is the cospan from $f_0$ to $f_1$ in the simplicial set $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ depicted informally in the diagram

$\xymatrix@C =50pt@R=50pt{ X_0 \ar [d]^{f_0} \ar [r]^-{\operatorname{id}} & X_0 \ar [d] & X_1 \ar [d]^{f_1} \ar [l]_{g} \\ Y_0 \ar [r]^-{h} & Y_1 & Y_1. \ar [l]_{\operatorname{id}} }$

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. It follows from Remark 8.2.6.12 that the morphism $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) )$ of Construction 8.2.6.11 factors through the simplicial subset $\operatorname{Cospan}^{L,R}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \subseteq \operatorname{Cospan}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) )$ appearing in the statement of Proposition 8.2.6.9. Unwinding the definitions, we obtain a morphism of couplings

8.55
$$\begin{gathered}\label{equation:Xi-to-cospan} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \Xi } \ar [d] & \operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}}) \ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r]^-{ \rho _{-} \times \rho _{+} } & \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}) \times \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) } \end{gathered}$$

where the vertical maps are the left fibrations of Proposition 8.1.1.11 and Lemma 8.2.6.10, and $\rho _{+}$ and $\rho _{-}$ are given by Construction 8.1.7.1 and Variant 8.1.7.14. By virtue of Remark 8.2.6.4, Proposition 8.2.6.9 is a consequence of the following more precise result:

Proposition 8.2.6.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the diagram (8.55) is an equivalence of couplings (in the sense of Exercise 8.2.2.7)

Proof. It follows from Proposition 8.1.7.6 that the inclusion maps

$\rho _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \hookrightarrow \operatorname{Cospan}^{\mathrm{iso}, \mathrm{all}}(\operatorname{\mathcal{C}}) \quad \quad \rho _{+}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}(\operatorname{\mathcal{C}})$

are equivalences of $\infty$-categories. By virtue of Corollary 5.1.7.15, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism $\Xi$ induces a homotopy equivalence of Kan complexes

$\Xi _{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Cospan}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ).$

We complete the proof by observing that $\Xi _{X,Y}$ fits into a commutative diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [d] \\ \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \ar [r]^-{ \Xi _{X,Y} } & \operatorname{Cospan}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) }$

where the left vertical map is the homotopy equivalence of Corollary 8.1.2.10, the right vertical map is the homotopy equivalence of Example 8.1.7.7, and the upper horizontal map is the homotopy equivalence of Proposition 4.6.5.10. $\square$

Corollary 8.2.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is an isomorphism if and only if it is universal with respect to the balanced coupling

$\lambda : \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}^{ \mathrm{iso}, \mathrm{all} }( \operatorname{\mathcal{C}}) \times \operatorname{Fun}^{ \mathrm{all}, \mathrm{iso} }( \operatorname{\mathcal{C}})$

of Proposition 8.2.6.9 (where we abuse notation by identifying $f$ with an object of the $\infty$-category $\operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) )$).

Proof. Let us abuse notation further by identifying $f$ with an object of the twisted arrow $\infty$-category $\operatorname{Tw}(\operatorname{\mathcal{C}})$, so that the comparison functor $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}^{L,R}( \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) )$ of Construction 8.2.6.11 satisfies $\Xi (f) = f$ (Remark 8.2.6.12). By virtue of Proposition 8.2.6.13 and Remark 8.3.2.8, we are reduced to showing that $f$ is an isomorphism if and only if it is universal with respect to the twisted arrow coupling $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$, which follows from Example 8.2.1.5. $\square$