Proposition 8.3.2.17. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a corepresentable profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$. The following conditions are equivalent:
- $(1)$
The profunctor $\mathscr {K}$ determines a fully faithful functor
\[ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, -). \]
- $(2)$
Let $X$ be an object of $\operatorname{\mathcal{C}}_{-}$ and let $Y$ be an object of $\operatorname{\mathcal{C}}_{+}$. Then every couniversal vertex $\eta \in \mathscr {K}(X,Y)$ is also universal.
Proof.
Choose an object $X \in \operatorname{\mathcal{C}}_{-}$. Since the functor $\mathscr {K}(X, -): \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is corepresentable, we can choose an object $Y \in \operatorname{\mathcal{C}}_{+}$ and a couniversal vertex $\eta \in \mathscr {K}(X,Y)$. We will show that the following conditions are equivalent:
- $(1_ X)$
For every object $X' \in \operatorname{\mathcal{C}}_{-}$, the profunctor $\mathscr {K}$ induces a homotopy equivalence
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}}( X, X' ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) }( \mathscr {K}(X, -), \mathscr {K}(X',-) ). \]
- $(2_ X)$
The vertex $\eta $ is universal.
Proposition 8.3.2.17 will then follow by allowing the triple $(X,Y,\eta )$ to vary.
Condition $(2_ X)$ is the assertion that, for each object $X' \in \operatorname{\mathcal{C}}_{-}$, the composite map
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}}( X, X' ) & \rightarrow & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) }( \mathscr {K}(X, -), \mathscr {K}(X', -) ) \\ & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \mathscr {K}(X,Y), \mathscr {K}(X', Y) ) \\ & \xrightarrow { \circ [\eta ] } & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {K}(X',Y) ) \\ & \simeq & \mathscr {K}(X',Y) \end{eqnarray*}
is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. The equivalence of this assertion with $(1_ X)$ follows immediately from Proposition 8.3.1.3.
$\square$