The Yoneda embedding of $\mathcal{C}$1 is the functor2
\[ {\text{よ}}_{\mathcal{C}}\colon \mathcal{C}\hookrightarrow \mathsf{PSh}(\mathcal{C}) \]
where
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Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, we have
\[ {\text{よ}}_{\mathcal{C}}(A) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h_{A}. \] -
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Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the action on morphisms
\[ {\text{よ}}_{\mathcal{C}|A,B}\colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)\to \operatorname {\mathrm{Nat}}(h_{A},h_{B}) \]of ${\text{よ}}_{\mathcal{C}}$ at $(A,B)$ is given by
\[ {\text{よ}}_{\mathcal{C}|A,B}(f)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h_{f} \]for each $f\in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)$, where $h_{f}$ is the representable natural transformation associated to $f$ of Definition 12.1.3.1.1.
- 1Further Terminology: Also called the covariant Yoneda embedding to distinguish it from the contravariant Yoneda embedding of Theorem 12.2.5.1.1.
- 2Further Notation: Also written $h_{(-)}$, or simply ${\text{よ}}$.
