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.
$\square$