
Tensor-hom adjunction

In mathematics, the tensor-hom adjunction is that the tensor product formula_1 and hom-functor formula_2 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 "F": "D" → "C" and "G": "C" → "D" 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 "C", 
is given on simple tensors of "Y"⊗"X" by
Likewise, 
For φ in "Hom"S("X", "Z"), 
is a right "S"-module homomorphism defined by
and therefore
The Ext and Tor functors.
The Hom functor formula_22 commutes with arbitrary limits, while the tensor product formula_23 functor commutes with arbitrary colimits that exist in their domain category. However, in general, formula_22 fails to commute with colimits, and formula_23 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.