# Kerodon

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Theorem 8.2.5.5 (Yoneda's Lemma for $\infty$-Categories). Let $\operatorname{\mathcal{C}}$ be a locally small $\infty$-category. Then the covariant and contravariant Yoneda embeddings

$h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$

are fully faithful functors, whose essential images are the full subcategories

$\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad \operatorname{Fun}^{\mathrm{corep}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$

spanned by the representable and corepresentable functors, respectively.