structural:expansion, structural:scholar_edition, linguistic:paraphrasing
Yoneda lemma

If X {\displaystyle X} is an object of a locally small category C {\displaystyle C}, then the assignment Y ↦ Hom C (X, Y) {\displaystyle Y\mapsto {\text{Hom}}_{C}(X,Y)} defines a covariant functor F X: C → Set {\displaystyle F_{X}:C\to {\textbf {Set}}}. This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of X {\displaystyle X}). The natural transformations from a representable functor to an arbitrary functor F: C → Set {\displaystyle F:C\to {\textbf {Set}}} are completely known and easy to describe; this is the content of the Yoneda lemma.