I was reading the book of McLane about Categories. In Yoneda's lemma he shows that a functor $F$ evaluated in a object $O$ can be written as the set of natural transformations between the representable functor of $O$ and $F$. My question is about a particular application of this theorem. Let $C$ be the category of finite rings with the limits of subcategories of $C$. Let $A$ be a finitely generated ring over the integers. Then evidently the tensor product with $A$ generates naturally a functor from $C$ to the category of sets. Let $F$ be this functor. My question is the following:

Is posible reconstruct $A$ as the set of natural transformacions from $F$ to itself?.

If this is not possible then:

Does there exist a category generated by finite objects and a functor $F$ such that we can reconstruct $A$ as a set of natural transformation from $F$ to itself?.

finitely-presentablerings. Then for any ring $A$, $Hom(-,A): {C'}^{op} \to Set$ is a functor from which $A$ can be reconstructed; in fact, the functor $Rng \to Fun({C'}^{op}, Set)$, $A \mapsto Hom(-,A)$ is fully faithful. Note thatfiniterings have certain restrictions, e.g. they are torsion, so they are not so great for probing rings of characteristic zero. $\endgroup$ – Tim Campion Jan 21 at 17:28