$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 8.4.2.5. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small. Then the profunctor
\[ \operatorname{ev}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad (X, \mathscr {F}) \mapsto \mathscr {F}(X) \]
is corepresentable by the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
Proof.
Let $\operatorname{Tw}(\operatorname{\mathcal{C}})$ denote the twisted arrow $\infty $-category of $\operatorname{\mathcal{C}}$, let $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ be the left fibration of Proposition 8.1.1.11, and let $\underline{ \Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ denote the constant functor $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$. Let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ denote the composition $\operatorname{ev}\circ (\operatorname{id}\times h_{\bullet })$, so that $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. We can therefore choose a natural transformation
\[ \alpha : \underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H} \circ \lambda = \operatorname{ev}\circ ( \operatorname{id}\times h_{\bullet } ) \circ \lambda \]
which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, in the sense of Definition 8.3.5.1. By virtue of Proposition 8.3.4.15, it will suffice to show that the natural transformation $\alpha $ also exhibits the profunctor $\operatorname{ev}$ as corepresented by the functor $h_{\bullet }$, in the sense of Variant 8.3.4.16. Fix an object $X \in \operatorname{\mathcal{C}}$, so that $\alpha $ carries the object $\operatorname{id}_{X} \in \operatorname{Tw}(\operatorname{\mathcal{C}})$ to a vertex $\eta \in \mathscr {H}(X,X) = \operatorname{ev}( X, h_ X)$. We wish to show that $\eta $ exhibits evaluation functor $\operatorname{ev}_{X}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ as corepresented by $h_{X}$. This follows from Proposition 8.3.1.3, since $\eta $ exhibits the functor $h_{X}$ as represented by $X$.
$\square$