essentially states that for every (locally small) category, objects in that category can be represented by presheaves, in a full and faithful manner. That is, for a presheaf P. Many common categories are, in fact, categories of pre-sheaves, and on closer inspection, prove to be categories of sheaves, and as such examples are commonly topological in nature, they can be seen to be topoi in general. The Yoneda lemma provides a point of leverage by which the topological structure of a category can be studied and understood. In terms of (co)end calculus Given two categories and with two functors , natural transformations between them can be written as the following end. For any functors and the following formulas are all formulations of the Yoneda lemma. Preadditive categories, rings and modules A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear; examples are categories of abelian groups or modules. In a preadditive category, there is both a "multiplication" and an "addition" of morphisms, which is why preadditive categories are viewed as generalizations of rings. Rings are preadditive categories with one object. The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category. The Yoneda lemma then yields the natural procedure to enlarge a preadditive category so that the enlarged version remains preadditive — in fact, the enlarged version is an abelian category, a much more powerful condition. In the case of a ring , the extended category is the category of all right modules over , and the statement of the Yoneda lemma reduces to the well-known isomorphism for all right modules over . Relationship to Cayley's theorem As stated above, the Yoneda lemma may be considered as a vast generalization of Cayley's theorem from group theory. To see this, let be a category with a single object such that every morphism is an isomorphism (i.e. a groupoid with one object). Then forms a group under the operation of composition, and any group can be realized as a category in this way. In this context, a covariant functor consists of a set and a group homomorphism ,