Proposition 8.5.5.5. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small, and let $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be the full subcategory spanned by those functors $\mathscr {F}$ for which the corepresentable functor $\operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( \mathscr {F}, \bullet )$ commutes with $\kappa $-small colimits. Then the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$.
Proof. To simplify the notation, set $\operatorname{\mathcal{D}}= \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. For each object $C \in \operatorname{\mathcal{C}}$, the representable functor $h_{C} \in \operatorname{\mathcal{D}}$ corepresents the functor
given by evaluation at $C$, which preserves $\kappa $-small colimits by virtue of Proposition 7.1.6.1. It follows that the covariant Yoneda embedding $h_{\bullet }$ factors through the subcategory $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{\mathcal{D}}$. Moreover, the functor $h_{\bullet }$ is fully faithful (Theorem 8.3.3.13).
The $\infty $-category $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits, and is therefore idempotent complete by virtue of Proposition 8.5.4.7. It follows from Corollary 8.5.1.14 that the full category $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{\mathcal{D}}$ is closed under the formation of retracts, and is therefore also idempotent complete (Proposition 8.5.4.6).
To complete the proof, it will suffice to show that every object $\mathscr {F} \in \widehat{\operatorname{\mathcal{C}}}$ is a retract of $h_{C}$ for some object $C \in \operatorname{\mathcal{C}}$. Applying Corollary 8.4.3.9, we deduce that $\mathscr {F}$ can be realized as the colimit of a diagram
where $\operatorname{\mathcal{K}}$ is an essentially $\kappa $-small $\infty $-category. Since the functor $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( \mathscr {F}, \bullet )$ preserves $\kappa $-small colimits, it follows that the identity map $\operatorname{id}_{\mathscr {F}} \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \mathscr {F}, \mathscr {F} )$ factors (up to homotopy) through $h_{ T(B) }$ for some $B$. In particular, $\mathscr {F}$ is a retract of the representable functor $h_{ T(B) }$. $\square$