Example 8.3.2.4 (From Profunctors to Couplings). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ (Definition 8.3.2.1). Applying the construction of Definition 5.6.2.1, we obtain an $\infty $-category $\int _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} } \mathscr {K}$ whose objects are triples $(X,Y,\eta )$ where $X$ is an object of $\operatorname{\mathcal{C}}_{-}$, $Y$ is an object of $\operatorname{\mathcal{C}}_{+}$, and $\eta $ is a vertex of the Kan complex $\mathscr {K}(X,Y)$. This $\infty $-category is equipped with a left fibration
given on objects by the construction $\lambda (X,Y,\eta ) = (X,Y)$ (see Example 5.6.2.9). The left fibration $\lambda $ is a coupling of $\operatorname{\mathcal{C}}_{+}$ with $\operatorname{\mathcal{C}}_{-}$, in the sense of Definition 8.2.0.1; we will refer to it as the coupling associated to the profunctor $\mathscr {K}$.