Composing this with the forgetful functor we have a contravariant functor from C to Set. Brown's representability theorem in algebraic topology says that this functor is represented by a CW-complex K(Z,n) called an Eilenberg–MacLane space. Let R be a commutative ring with identity, and let R-Mod be the category of R-modules. If M and N are unitary modules over R, there is a covariant functor B: R-Mod → Set which assigns to each R-module P the set of R-bilinear maps M × N → P and to each R-module homomorphism f : P → Q the function B(f) : B(P) → B(Q) which sends each bilinear map g : M × N → P to the bilinear map f∘g : M × N→Q. The functor B is represented by the R-module M ⊗R N. Properties Uniqueness Representations of functors are unique up to a unique isomorphism. That is, if (A1,Φ1) and (A2,Φ2) represent the same functor, then there exists a unique isomorphism φ : A1 → A2 such that as natural isomorphisms from Hom(A2,–) to Hom(A1,–). This fact follows easily from Yoneda's lemma. Stated in terms of universal elements: if (A1,u1) and (A2,u2) represent the same functor, then there exists a unique isomorphism φ : A1 → A2 such that Preservation of limits Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covariant) representable functors preserve all limits. It follows that any functor which fails to preserve some limit is not representable. Contravariant representable functors take colimits to limits. Left adjoint Any functor K : C → Set with a left adjoint F : Set → C is represented by (FX, ηX(•)) where X = {•} is a singleton set and η is the unit of the adjunction. Conversely, if K is represented by a pair (A, u) and all small copowers of A exist in C then K has a left adjoint F which sends each set I to the Ith copower of A. Therefore, if C is a category with all small copowers, a functor K : C → Set is representable if and only if it has a left adjoint. Relation to universal morphisms and adjoints The categorical notions of universal morphisms and adjoint functors can both be expressed using representable functors. Let G : D → C be a functor and let X be an object of C. Then (A,φ)