Tensor-hom adjunction

In mathematics, the tensor-hom adjunction is that the tensor product and Hom functors ${\displaystyle -\otimes X}$ and ${\displaystyle \operatorname {Hom} (X,-)}$ form an adjoint pair:

This is made more precise below. The order "tensor-hom adjunction" is because tensor is the left adjoint, while hom is the right adjoint.

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.

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