8.2.2 Morphisms of Couplings
We begin by introducing a companion of Definition 8.2.0.1.
Definition 8.2.2.1. 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. A morphism of couplings from $\lambda $ to $\mu $ is a triple of functors
\[ F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{D}}_{-} \quad \quad F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}\quad \quad F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{D}}_{+} \]
for which the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{\lambda } & \operatorname{\mathcal{D}}\ar [d]^{\mu } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{F^{\operatorname{op}}_{-} \times F_{+}} & \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} } \]
is commutative. Note that such diagrams can be identified with the vertices of a simplicial set $\operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, defined by the formula
\[ \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) = \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} ) } \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ). \]
Proposition 8.2.2.2. 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. Then the projection maps
\[ \Phi _{-}: \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \quad \quad \Phi _{+}: \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \]
induce a left fibration
\[ ( \Phi _{-}, \Phi _{+}): \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ). \]
Proof.
By construction, there is a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r] \ar [d]^{ ( \Phi _{-}, \Phi _{+})} & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [d]^{ \mu \circ } \\ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \ar [r]^-{ \circ \lambda } & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} ). } \]
It will therefore suffice to show that the right vertical map is a left fibration (Remark 4.2.1.8), which follows from our assumption that $\mu $ is a left fibration (Corollary 4.2.5.2).
$\square$
Corollary 8.2.2.5. 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. Then the simplicial set $\operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is an $\infty $-category.
Proof.
Combine Proposition 8.2.2.2 with Remark 4.2.1.4.
$\square$
Example 8.2.2.6. The canonical isomorphism $\lambda : \Delta ^0 \xrightarrow {\sim } (\Delta ^0)^{\operatorname{op}} \times \Delta ^0$ can be regarded as a coupling of the $0$-simplex $\Delta ^0$ with itself. For every coupling of $\infty $-categories $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+}$, the $\infty $-category $\operatorname{Fun}_{\pm }( \Delta ^0, \operatorname{\mathcal{D}})$ can be identified with the $\infty $-category $\operatorname{\mathcal{D}}$.
Exercise 8.2.2.7. 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, and suppose we are given a morphism of couplings
8.30
\begin{equation} \begin{gathered}\label{equation:equivalence-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}}_{+} } \end{gathered} \end{equation}
Show that the following conditions are equivalent:
The functors $F_{-}$, $F$, and $F_{+}$ are equivalences of $\infty $-categories.
There exists a morphism of couplings $(G_{-}, G, G_{+}) \in \operatorname{Fun}_{\pm }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ which is a homotopy inverse to $(F_{-}, F, F_{+} )$, in the sense that the compositions $( G_{-} \circ F_{-}, G \circ F, G_{+} \circ F_{+} )$ and $(F_{-} \circ G_{-}, F \circ G, F_{+} \circ G_{+} )$ are isomorphic to $( \operatorname{id}_{\operatorname{\mathcal{C}}_{-}}, \operatorname{id}_{\operatorname{\mathcal{C}}}, \operatorname{id}_{\operatorname{\mathcal{C}}_{+}})$ and $( \operatorname{id}_{ \operatorname{\mathcal{D}}_{-}}, \operatorname{id}_{\operatorname{\mathcal{D}}}, \operatorname{id}_{ \operatorname{\mathcal{D}}_{+} } )$ as objects of the $\infty $-categories $\operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ and $\operatorname{Fun}_{\pm }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, respectively.
If these conditions are satisfied, we will say that the diagram (8.30) is an equivalence of couplings.
See Corollaries 4.6.7.21 and 4.6.7.20.
Beware that, in the situation of Definition 8.2.2.1, the $\infty $-category $\operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ depends not only on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, but also on the left fibrations $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-} \times \operatorname{\mathcal{C}}_{+}$ and $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-} \times \operatorname{\mathcal{D}}_{+}$. Our goal in this section is to show that, nevertheless, it can often be identified with a full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (Proposition 8.2.2.9).
Proposition 8.2.2.9. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ and $\mu = (\mu _{-}, \mu _{+}): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+}$ be couplings of $\infty $-categories. Let $\operatorname{\mathcal{E}}_{-} \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ be the full subcategory spanned by those functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carry $\lambda _{+}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ to $\mu _{+}$-cocartesian morphisms of $\operatorname{\mathcal{D}}$, define $\operatorname{\mathcal{E}}_{+} \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ similarly, and set $\operatorname{\mathcal{E}}_{\pm } = \operatorname{\mathcal{E}}_{-} \cap \operatorname{\mathcal{E}}_{+}$. Then:
- $(1)$
Suppose that, for every object $C_{-} \in \operatorname{\mathcal{C}}_{-}$, the $\infty $-category $\{ C_{-} \} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}} \operatorname{\mathcal{C}}$ is weakly contractible. Then the forgetful functor
\[ \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{E}}_{-} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} )} \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \]
is an equivalence of $\infty $-categories.
- $(2)$
Suppose that, for every object $C_{+} \in \operatorname{\mathcal{C}}_{+}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}_{+}} \{ C_{+} \} $ is weakly contractible. Then the forgetful functor
\[ \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-})^{\operatorname{op}} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )} \operatorname{\mathcal{E}}_{+} \]
is an equivalence of $\infty $-categories.
- $(3)$
If the hypotheses of both $(1)$ and $(2)$ are satisfied, then the forgetful functor $\operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{E}}_{\pm }$ is an equivalence of $\infty $-categories.
Proof.
We first prove $(1)$. Let $W$ be the collection of all $\lambda _{+}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$. Note that a morphism $u$ of $\operatorname{\mathcal{C}}$ belongs to $W$ if and only if $\lambda _{-}(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ (Proposition 8.2.1.7). Suppose that, for every object $C_{-} \in \operatorname{\mathcal{C}}_{-}$, the $\infty $-category $\{ C_{-} \} \times _{\operatorname{\mathcal{C}}_{-}} \operatorname{\mathcal{C}}$ is weakly contractible. Applying Corollary 6.3.5.3, we deduce that the functor $\lambda _{-}$ exhibits $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. It follows that precomposition with $\lambda _{-}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )$, where $\operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )$ spanned by those functors which carry each element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}^{\operatorname{op}}_{-}$ (Notation 6.3.1.1). We have a commutative diagram of $\infty $-categories
8.31
\begin{equation} \begin{gathered}\label{equation:collapse-left-universal} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \ar [d]^{\circ \lambda _{-}} \\ \operatorname{\mathcal{E}}_{-} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} ) } \operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ) \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+}) } \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \ar [r]^-{\theta } & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ) } \end{gathered} \end{equation}
in which both squares are pullbacks. To prove $(1)$, it will suffice to show that the upper square is a categorical pullback diagram (Proposition 4.5.2.21). In fact, we will show that $\theta $ is an isofibration, so that both squares are categorical pullback diagrams (Corollary 4.5.2.27). This follows by observing that $\theta $ factors as a composition
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} )} \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \xrightarrow {\theta '} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ) \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \xrightarrow {\theta ''} \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ), \]
where $\theta '$ is a pullback of the composition map $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow { \mu \circ } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} )$ (hence a left fibration by virtue of Corollary 4.2.5.2) and $\theta ''$ is a pullback of the projection map $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \rightarrow \Delta ^0$. This completes the proof of assertion $(1)$.
Assertion $(2)$ follows by a similar argument. We now prove $(3)$. Suppose that $\lambda $ satisfies the hypotheses of both $(1)$ and $(2)$; we wish to prove that the forgetful functor $T: \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{E}}_{\pm }$ is an equivalence of $\infty $-categories. Note that $T$ factors as a composition
\[ \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow {T'} \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )} \operatorname{\mathcal{E}}_{+} \xrightarrow {T''} \operatorname{\mathcal{E}}_{\pm }, \]
where $T'$ is an equivalence of $\infty $-categories by virtue of $(2)$. It will therefore suffice to show that $T''$ is an equivalence of $\infty $-categories. We have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )} \operatorname{\mathcal{E}}_{+} \ar [d]^{T''} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \ar [d]^{ \circ \lambda _{-} } \\ \operatorname{\mathcal{E}}_{\pm } \ar [r]^-{\mu _{-} \circ } \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ) \ar [d] \\ \operatorname{\mathcal{E}}_{+} \ar [r]^-{\rho } & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ), } \]
where both squares are pullbacks and the upper right vertical map is an equivalence of $\infty $-categories. It will therefore suffice to show that the upper square is a categorical pullback diagram (Proposition 4.5.2.21). In fact, we claim that $\rho $ is an isofibration, so that both squares are categorical pullback diagrams (Corollary 4.5.2.27). This follows by observing that $\rho $ is the restriction of the map $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow { \mu _{-} \circ } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )$ (which is a cocartesian fibration by virtue of Proposition 8.2.1.7 and Theorem 5.2.1.1) to a replete subcategory $\operatorname{\mathcal{E}}_{+} \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
$\square$
Corollary 8.2.2.10. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ and $\mu = (\mu _{-}, \mu _{+}): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+}$ be couplings of $\infty $-categories. Fix a functor $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{D}}_{+}$. If $\lambda $ is corepresentable, then the forgetful functor
\[ \operatorname{Fun}_{ \pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} )} \{ F_{+} \} \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{+} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]
is fully faithful, and its essential image is the full subcategory $\operatorname{Fun}^{0}_{ / \operatorname{\mathcal{D}}_{+} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{+} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which carry $\lambda _{+}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ to $\mu _{+}$-cocartesian morphisms of $\operatorname{\mathcal{D}}$.
Proof.
Let $\operatorname{\mathcal{E}}_{-} \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ be the full subcategory defined in Proposition 8.2.2.9. We then have a commutative diagram of $\infty $-categories
8.32
\begin{equation} \begin{gathered}\label{equation:collapse-left-universal2} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) } \{ F_{+} \} \ar [r] \ar [d] & \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}^{0}_{ / \operatorname{\mathcal{D}}_{+} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{-} \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} ) } \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \ar [d] \\ \{ F_{+} \} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ), } \end{gathered} \end{equation}
where both squares are pullback diagrams. Note that the vertical map on the lower right is a pullback of the functor $\operatorname{\mathcal{E}}_{-} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} )$ obtained by restricting the cocartesian fibration $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow {\mu _{+} \circ } \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+})$ (see Proposition 8.2.1.7 and Theorem 5.2.1.1) to the replete subcategory $\operatorname{\mathcal{E}}_{-} \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, and is therefore an isofibration. Moreover, the right vertical composition $\operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} )$ is a cocartesian fibration (see Proposition 8.2.1.7 and Remark 8.2.2.3), and therefore an isofibration. It follows that the bottom square and outer rectangle of (8.32) are categorical pullback diagrams (Corollary 4.5.2.27), so that the upper square is also a categorical pullback diagram (Proposition 4.5.2.18). Our assumption that $\lambda $ is corepresentable guarantees that for each object $X_{-} \in \operatorname{\mathcal{C}}_{-}$, the fiber $\{ X_{-} \} \times _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{C}}$ has an initial object, and is therefore weakly contractible. Applying Proposition 8.2.2.9, we deduce that the vertical map on the upper right is an equivalence of $\infty $-categories. Invoking Proposition 4.5.2.21, we conclude that the forgetful functor $\operatorname{Fun}_{ \pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} )} \{ F_{+} \} \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{+}}^{0}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is also an equivalence of $\infty $-categories.
$\square$
We can now formulate the main result of this section.
Theorem 8.2.2.11. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ be a representable coupling of $\infty $-categories and let $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ be a corepresentable coupling of $\infty $-categories. Then the functor coupling
\[ \Phi = ( \Phi _{-}, \Phi _{+}): \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \]
is corepresentable. Moreover, an object $(F_{-}, F, F_{+}) \in \operatorname{Fun}_{\pm }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is couniversal if and only if the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ carries universal objects of $\operatorname{\mathcal{C}}$ to couniversal objects of $\operatorname{\mathcal{D}}$.
Proof.
Fix a functor $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{D}}_{-}$; we wish to show that it can be extended to a couniversal object $( F_{-}, F, F_{+} ) \in \operatorname{Fun})_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Set $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times _{\operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$. Then projection onto the first factor determines a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ (Proposition 8.2.1.7). Let $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ denote the full subcategory of $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by those functors which carry $\lambda _{-}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ to $U$-cocartesian of $\operatorname{\mathcal{E}}$ (Notation 5.3.1.10). Corollary 8.2.2.10 guarantees that the forgetful functor
\[ \{ F_{-} \} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]
is an equivalence of $\infty $-categories. Theorem 8.2.2.11 can therefore be restated as follows:
- $(1)$
The $\infty $-category $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ has an initial object.
- $(2)$
An object $F \in \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is initial if and only if, for universal object $C \in \operatorname{\mathcal{C}}$, the image $F( C)$ is an initial object of the $\infty $-category
\[ \operatorname{\mathcal{E}}_{ \lambda _{-}(C)} \simeq \{ F_{-}(\lambda _{-}(C)) \} \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}. \]
Our assumption that $\mu $ is corepresentable guarantees that, for each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{\lambda _{-}(C)}$ has an initial object. Consequently, assertions $(1)$ and $(2)$ follow from (the dual of) Proposition 8.2.1.8.
$\square$
Example 8.2.2.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then the commutative diagram
8.33
\begin{equation} \begin{gathered}\label{equation:example:easy-corepresentable-hom} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \operatorname{Tw}(F) } \ar [d] & \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}}} \end{gathered} \end{equation}
is a morphism of couplings. It follows from Theorem 8.2.2.11 that this morphism is initial when viewed as an object of the $\infty $-category $\{ F \} \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$.
Corollary 8.2.2.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and suppose we are given a pair of functors $F_{-}, F_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Then $F_{-}$ and $F_{+}$ are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$) if and only if there exists a morphism of couplings
8.34
\begin{equation} \begin{gathered}\label{equation:isomorphism-of-functors-twisted-arrow} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \widetilde{F} } \ar [d] & \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}}} \end{gathered} \end{equation}
having the property that the functor $\widetilde{F}$ carries isomorphisms of $\operatorname{\mathcal{C}}$ (regarded as objects of $\operatorname{Tw}(\operatorname{\mathcal{C}})$) to isomorphisms of $\operatorname{\mathcal{D}}$ (regarded as objects of $\operatorname{Tw}(\operatorname{\mathcal{D}})$).
Proof.
Suppose first that there exists an isomorphism of functors $\alpha : F_{-} \rightarrow F_{+}$. Since the projection map $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$ is an isofibration, we can use Corollary 4.4.5.6 to lift the natural transformation
\[ (\operatorname{id}\times \alpha ): F_{-}^{\operatorname{op}} \times F_{+} \rightarrow F_{-}^{\operatorname{op}} \times F_{-} \]
to an isomorphism $\widetilde{F} \rightarrow \operatorname{Tw}(F_{-})$ in the $\infty $-category $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$, so that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \widetilde{F} } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r]^-{ F_{-}^{\operatorname{op}} \times F_{+} } & \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}} \]
where $\widetilde{F}$ carries isomorphisms of $\operatorname{\mathcal{C}}$ to isomorphisms of $\operatorname{\mathcal{D}}$.
We now prove the converse. Suppose we are given a commutative diagram (8.34), where $\widetilde{F}$ carries isomorphisms of $\operatorname{\mathcal{C}}$ to isomorphisms of $\operatorname{\mathcal{D}}$. Applying Theorem 8.2.2.11, we deduce that the triple $( F_{-}, \widetilde{F}, F_{+} )$ is initial when viewed as an object of the $\infty $-category $\{ F_{-} \} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$. Applying Example 8.2.2.12 (and Corollary 4.6.7.15), we deduce that $( F_{-}, \widetilde{F}, F_{+} )$ is isomorphic to $( F_{-}, \operatorname{Tw}(F_{-}), F_{-} )$ as an object of the $\infty $-category $\{ F_{-} \} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$. In particular, $F_{+}$ is isomorphic to $F_{-}$ as an object of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
$\square$