Proposition 8.2.5.2 (045T)

Proposition 8.2.5.2. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ be coupling of $\infty $-categories which is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$. The following conditions are equivalent:

$(1)$: The coupling $\lambda $ is corepresentable.
$(2)$: The functor $G$ admits a left adjoint.

Proof. By virtue of the criterion of Corollary 6.2.4.2, it will suffice to show that for each object $X \in \operatorname{\mathcal{C}}_{-}$, the following conditions are equivalent:

$(1_ X)$: There exists a couniversal object $\widetilde{X} \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{-}( \widetilde{X} ) = X$.
$(2_ X)$: The $\infty $-category $( \operatorname{\mathcal{C}}_{-} )_{X/} \times _{ \operatorname{\mathcal{C}}_{-} } \operatorname{\mathcal{C}}_{+}$ has an initial object.

Note that Proposition 8.1.2.9 supplies an equivalence $(\operatorname{\mathcal{C}}_{-})_{X/} \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} } \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} )$ of $\infty $-categories which are left-fibered over $\operatorname{\mathcal{C}}_{-}$. Restricting along the functor $G$, we obtain an equivalence of $\infty $-categories

\[ ( \operatorname{\mathcal{C}}_{-} )_{X/} \times _{ \operatorname{\mathcal{C}}_{-} } \operatorname{\mathcal{C}}_{+} \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}_{-} }^{\operatorname{op}} \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{ \operatorname{\mathcal{C}}_{-} } \operatorname{\mathcal{C}}_{+}. \]

Since $\lambda $ is representable by $G$, there exists a categorical pullback square

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

which induces an equivalence of $\infty $-categories

\[ \{ X\} \times _{\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}} \operatorname{\mathcal{C}}\rightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}_{-} }^{\operatorname{op}} \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{ \operatorname{\mathcal{C}}_{-} } \operatorname{\mathcal{C}}_{+}. \]

The equivalence of $(1_ X)$ and $(2_ X)$ now follows from Corollary 4.6.7.21. $\square$