Why can't we use Yoneda Lemma to get a representation theorem for Rings? I'm new to category theory.
I am trying to understand how Yoneda Lemma is a generalization of representations theorems in algebra.
Cayley theorem can be interpreted as a instance for the case where the category $\mathcal{C}$ is the one with only one element $*$ and  arrows are iso arrows that correspond to the elements of $G$ (i.e., it is a grupoid. I am using this wiki-page as reference) . From the Yoneda Lemma we can establish the Cayley theorem.
But there is another result from algebra that says that there isn't a representation theorem for Rings in general. (I don't understand why this is the case, I just heard it). There is only a representation theorem if we restrict to the context of Boolean algebras (the Stone Theorem).
Why we can't just apply Yoneda lemma in some way to get a general result for Ring theory? (Not the Stone theorem) I mean, in the context of groups we've used the grupoid category. Why we can't consider something like it in Rings and apply the Yoneda? Some category like a ``ringoid'' (sorry for this word).
Maybe there is to much happening here, but I think some intuition of this might help understand better the nature of Yoneda Lemma.
Edit: the discussion in the comments and in this link pointed me that the result I mentioned is false: there is a Cayley on Rings. I will not edit the original text question. So now I think the question is:

How do we get this version of Cayley on Rings with Yoneda?

I still very confused with all this new information, and I didn't understand the answer there.
 A: A preadditive category is a category $\mathcal{A}$ together with the structure of an abelian group of each set $\mathcal{A}(A, B)$, in such a way that that all composition functions
$$
  \mathrm{Hom}_{\mathcal{A}}(A, B)
  ×
  \mathrm{Hom}_{\mathcal{A}}(B, C)
  \longrightarrow
  \mathrm{Hom}_{\mathcal{A}}(A, C) \,,
  \quad
  (f, g) \longmapsto g ∘ f
$$
are $ℤ$-bilinear. More explicitly,
$$
  g ∘ (f_1 + f_2) = g ∘ f_1 + g ∘ f_2 \,,
  \quad
  (g_1 + g_2) ∘ f = g_1 ∘ f + g_2 ∘ f \,.
$$
For any object $A$ in a preadditive category $\mathcal{A}$, the set $\mathrm{End}_{\mathcal{A}}(A)$ becomes a ring:
its underlying abelian group is $\mathrm{Hom}_{\mathcal{A}}(A, A)$, and its multiplication is given by composition of endomorphisms.
There are two important examples of preadditive categories are the following:

*

*The category $\mathbf{Ab}$ of abelian groups.


*Every ring $R$ can be regarded as a preadditive category $\mathcal{R}$ consisting of only a single object $\ast$, with $\mathrm{End}_{\mathcal{R}}(A) = R$.
For any preadditive category $\mathcal{A}$, its opposite category $\mathcal{A}^{\mathrm{op}}$ is again preadditive.

A functor $F \colon \mathcal{A} \to \mathcal{B}$ between preadditive categories $\mathcal{A}$ and $\mathcal{B}$ is called additive if for any two objects $A$ and $B$, the map
$$
  \mathrm{Hom}_{\mathcal{A}}(A, B)
  \xrightarrow{\enspace F \enspace}
  \mathrm{Hom}_{\mathcal{B}}( F(A), F(B) )
$$
is a homomorphism of groups.
If $F$ is such an additive functor, then for every object $A$ of $\mathcal{A}$, the map
$$
  \mathrm{End}_{\mathcal{A}}(A)
  \xrightarrow{\enspace F \enspace}
  \mathrm{End}_{\mathcal{B}}(F(A))
$$
is a homomorphism of rings.
An important example of additive functors are represented functors:
for every object $A$ of a preadditive category $\mathcal{A}$, both
$$
  \mathrm{Hom}_{\mathcal{A}}(A, -)
  \colon
  \mathcal{A} \longrightarrow \mathbf{Ab}
$$
and
$$
  \mathrm{Hom}_{\mathcal{A}}(-, A)
  \colon
  \mathcal{A}^{\mathrm{op}} \longrightarrow \mathbf{Ab}
$$
are additive.

Given two functors
$$
  F, G \colon \mathcal{A} \longrightarrow \mathcal{B}
$$
with $\mathcal{B}$ preadditive, the set of natural transformation from $F$ to $G$, denoted by $\mathrm{Nat}(F, G)$, becomes an abelian group with addition given by
$$
  (α + β)_A ≔ α_α + β_A
$$
for every two natural transformations $α$ and $β$ and every object $A$ of $\mathcal{A}$.
It follows that the functor category $[\mathcal{A}, \mathcal{B}]$ inherits a preadditive structure from $\mathcal{B}$.

One can now state a version of Yoneda’s lemma for preadditive categories:

(Yoneda’s lemma for preadditive categories.)
Let $\mathcal{A}$ be a preadditive category, let $F$ be an additive functor from $\mathcal{A}^{\mathrm{op}}$ to $\mathbf{Ab}$, and let $A$ be an object of $\mathcal{A}$.
The map
$$
  \mathrm{Nat}( \mathrm{Hom}_{\mathcal{A}}(-, A), F )
  \longrightarrow
  F(A) \,,
  \quad
  α \longmapsto α_A(\mathrm{id}_A)
$$
is an isomorphism of abelian groups that is natural in both $A$ and $F$.

As a consequence, one gets a Yoneda embedding for preadditive categories.

(Yoneda embedding for preadditive categories)
Let $\mathcal{A}$ be a preadditive category.
Then
$$
  \mathcal{A}
  \longrightarrow
  [\mathcal{A}^{\mathrm{op}}, \mathbf{Ab}] \,,
  \quad
  A
  \longmapsto
  \mathrm{Hom}(-, A)
$$
is an additive embedding of categories.

As a consequence, one find that for every object $A$ of a preadditive category $\mathcal{A}$, the map
$$
  \mathrm{End}_{\mathcal{A}}(A)
  \longrightarrow
  \mathrm{End}_{[\mathcal{A}^{\mathrm{op}}, \mathbf{Ab}]}( 
    \mathrm{Hom}_{\mathcal{A}}(-, A)
  )
$$
is an isomorphism of rings.

Given a ring $R$, we can form the corresponding preadditive category $\mathcal{R}$ consisting of only a single element $\ast$ with $\mathrm{End}_{\mathcal{R}}(\ast) = R$.
A functor from $\mathcal{R}^{\mathrm{op}}$ to $\mathbf{Ab}$ is then “the same” as a right $R$-module.
The right $R$-module corresponding to the represented functor $\mathcal{R}(-, \ast)$ is just $R$ itself.
Yoneda’s lemma gives us isomorphisms of rings
$$
  R
  =
  \mathrm{End}_{\mathcal{R}}(\ast)
  ≅
  \mathrm{End}_{[\mathcal{R}^{\mathrm{op}}, \mathbf{Ab}]}( 
    \mathrm{Hom}_{\mathcal{R}}(-, \ast)
  )
  ≅
  \mathrm{End}_{\mathrm{Mod}\text{-}R}(R) \,.
$$
This overall isomorphism sends any element $r$ of $R$ to the map
$$
  R \longrightarrow R \,,
  \quad
  x \longmapsto r x \,,
$$
which is indeed a homomorphism of right $R$-modules.
The ring $\mathrm{End}_{\mathrm{Mod}\text{-}R}(R)$ is a subring of $\mathrm{End}_{ℤ}(R)$.
Therefore, every ring $R$ can be realized as a subring of $\mathrm{End}_ℤ(A)$ for some abelian group $A$.
This is Cayley’s theorem for rings.
