morphism a value of . Contravariant version There is a contravariant version of Yoneda's lemma, which concerns contravariant functors from to . This version involves the contravariant hom-functor which sends to the hom-set . Given an arbitrary contravariant functor from to , Yoneda's lemma asserts that Naming conventions The use of for the covariant hom-functor and for the contravariant hom-functor is not completely standard. Many texts and articles either use the opposite convention or completely unrelated symbols for these two functors. However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article. The mnemonic "falling into something" can be helpful in remembering that is the covariant hom-functor. When the letter is falling (i.e. a subscript), assigns to an object the morphisms from into . Proof Since is a natural transformation, we have the following commutative diagram: This diagram shows that the natural transformation is completely determined by since for each morphism one has Moreover, any element defines a natural transformation in this way. The proof in the contravariant case is completely analogous. The Yoneda embedding An important special case of Yoneda's lemma is when the functor from to is another hom-functor . In this case, the covariant version of Yoneda's lemma states that That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in the reverse direction) between the associated objects. Given a morphism the associated natural transformation is denoted . Mapping each object in to its associated hom-functor and each morphism to the corresponding natural transformation determines a contravariant functor from to , the functor category of all (covariant) functors from to . One can interpret as a covariant functor: The meaning of Yoneda's lemma in this setting is that the functor is fully faithful, and therefore gives an embedding of in the category of functors to . The collection of all functors is a subcategory of . Therefore, Yoneda embedding implies that the category is isomorphic to the category . The contravariant version of Yoneda's lemma states that Therefore, gives rise to a covariant functor from to the category of contravariant functors to : Yoneda's lemma then states that any locally small category can be embedded in the category of contravariant functors from to via . This is called the Yoneda embedding. The Yoneda embedding is sometimes denoted by よ, the Hiragana kana Yo. Representable functor The Yoneda embedding