Kerodon

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Definition 8.2.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let

\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto h_ Y \]

be a functor. We say that $h_{\bullet }$ is a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ if the construction $(X,Y) \mapsto h_{Y}(X)$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, in the sense of Definition 8.2.3.2. Similarly, we say that a functor

\[ h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto h^{X} \]

is a contravariant Yoneda embedding for $\operatorname{\mathcal{C}}$ if the construction $(X,Y) \mapsto h^{X}(Y)$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.