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We can read in the nLab (here), that for $C$ a locally small category, the Yoneda Embedding

$$ Y : C \to [C^{op}, Set] $$

is the image of the $hom$ functor

$$ Hom : C^{op} \times C \to Set $$

under the adjunction

$$ Hom(C^{op} \times C , Set) \simeq Hom(C, [C^{op}, Set]) $$

in the closed symmetric monoidal category $Cat$.

But, in a general symmetric monoidal category (according to this) we had

$$ Hom_C(a \otimes b, c) \simeq Hom_C(a, [b,c]) $$

And in the category $Cat$ particularly, to get the right hand side of the former adjunction, $c$ would be $Set$, $b$ has to be $C^{op}$, and $a$ then $C$ (with the tensor product being the product of categories, and the square bracket the hom functor). Then we would have:

$$ Hom_{Cat}(C \times C^{op}, Set) \simeq Hom_{Cat}(C, hom(C^{op}, Set)), $$

but then, in the left hand side, the order of the "ops" result wrong, it should be $C^{op} \times C$ instead of $C \times C^{op}$. Can you help me find what is wrong?

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1 Answer 1

up vote 2 down vote accepted

The monoidal product is the cartesian one, that means that it is symmetric: there's a natural isomorphism $X \times Y \stackrel{i}{\cong} Y \times X$ for every pair $X,Y \in \mathbf {Cat}$.

So to be correct the yoneda embedding is obtained as the image of the $\hom$-functor via the natural isomorphism $$\hom_{\mathbf {Cat}}(\mathbf C^\text{op} \times \mathbf C,\mathbf{Set}) \stackrel{i^*}{\cong} \hom_{\mathbf {Cat}}(\mathbf C \times \mathbf C^\text{op},\mathbf {Set}) \cong \hom_{\mathbf {Cat}}(\mathbf C, \hom(\mathbf C^\text{op},\mathbf {Set}))$$

Hope this helps.

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