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# Yoneda lemma

If $\mathcal{C}$ is a category, write $\hat{\mathcal{C}}$ for the category of contravariant functors from $\mathcal{C}$ to ${\bf Sets}$, the category of sets. The morphisms in $\hat{\mathcal{C}}$ are natural transformations of functors.

(To avoid set theoretical concerns, one can take a universe $\mathcal{U}$ and take all categories to be $\mathcal{U}$-small.)

For any object $X$ of $\mathcal{C}$, $h_{X}={\rm Hom}(-,X)$ is a contravariant functor from $\mathcal{C}$ to ${\bf Sets}$, and therefore is an object of $\hat{\mathcal{C}}$.

*Yoneda Lemma* says that $X\mapsto h_{X}$ is a covariant functor $\mathcal{C}\to\hat{\mathcal{C}}$, which embeds $\mathcal{C}$ faithfully as a full subcategory of $\hat{\mathcal{C}}$. This embedding is called the *Yoneda embedding*.

## Mathematics Subject Classification

18A25*no label found*

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## Comments

## what about the yoneda product?

i know they are different, err different fields even, but i am trying to get a good formal definitiion, thanx fer yer time