Kerodon

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Example 8.4.5.2 (Representable Functors are Atomic). Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category. Then every representable functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is atomic when regarded as an object of the $\infty $-category $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. To see this, suppose that $\mathscr {F}$ is representable by an object $C \in \operatorname{\mathcal{C}}$. Using Remark 8.3.1.5, we see that $\mathscr {F}$ corepresents the evaluation functor

\[ \operatorname{ev}_{C}: \widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {G} \mapsto \mathscr {G}(C), \]

and therefore preserves small colimits by virtue of Proposition 7.1.6.1.