In mathematics, the tensor-hom adjunction is that the tensor product - \otimes X and hom-functor form an adjoint pair: This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint.

General statement

Say R and S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules): Fix an (R,S)-bimodule X and define functors and as follows: Then F is left adjoint to G. This means there is a natural isomorphism This is actually an isomorphism of abelian groups. More precisely, if Y is an (A,R)-bimodule and Z is a (B,S)-bimodule, then this is an isomorphism of (B,A)-bimodules. This is one of the motivating examples of the structure in a closed bicategory.

Counit and unit

Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations. Using the notation from the previous section, the counit has components given by evaluation: For The components of the unit are defined as follows: For y in Y, is a right S-module homomorphism given by The counit and unit equations can now be explicitly verified. For Y in \mathcal{D}, is given on simple tensors of Y \otimes X by Likewise, For \phi in , is a right S-module homomorphism defined by and therefore

The Ext and Tor functors

The Hom functor \hom(X,-) commutes with arbitrary limits, while the tensor product -\otimes X functor commutes with arbitrary colimits that exist in their domain category. However, in general, \hom(X,-) fails to commute with colimits, and -\otimes X fails to commute with limits; this failure occurs even among finite limits or colimits. This failure to preserve short exact sequences motivates the definition of the Ext functor and the Tor functor.