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?.