# Tensor-hom adjunction

In mathematics, the tensor-hom adjunction is that the tensor product $-\otimes X$ and hom-functor $\operatorname {Hom} (X,-)$ form an adjoint pair:
    $\operatorname {Hom} (Y\otimes X,Z)\cong \operatorname {Hom} (Y,\operatorname {Hom} (X,Z)).$
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):
    ${\mathcal {C}}=\mathrm {Mod} _{S}\quad {\text{and}}\quad {\mathcal {D}}=\mathrm {Mod} _{R}.$
Fix an $(R,S)$-bimodule $X$ and define functors $F\colon {\mathcal {D}}\rightarrow {\mathcal {C}}$ and $G\colon {\mathcal {C}}\rightarrow {\mathcal {D}}$ as follows:
    $F(Y)=Y\otimes _{R}X\quad {\text{for }}Y\in {\mathcal {D}}$
    $G(Z)=\operatorname {Hom} _{S}(X,Z)\quad {\text{for }}Z\in {\mathcal {C}}$
Then $F$ is left adjoint to $G$. This means there is a natural isomorphism
    $\operatorname {Hom} _{S}(Y\otimes _{R}X,Z)\cong \operatorname {Hom} _{R}(Y,\operatorname {Hom} _{S}(X,Z)).$
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
    $\varepsilon :FG\to 1_{\mathcal {C}}$
has components
    $\varepsilon _{Z}:\operatorname {Hom} _{S}(X,Z)\otimes _{R}X\to Z$
given by evaluation: For
    $\phi \in \operatorname {Hom} _{S}(X,Z)\quad {\text{and}}\quad x\in X,$
    $\varepsilon (\phi \otimes x)=\phi (x).$
The components of the unit
    $\eta :1_{\mathcal {D}}\to GF$
    $\eta _{Y}:Y\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)$
are defined as follows: For $y$ in $Y$,
    $\eta _{Y}(y)\in \operatorname {Hom} _{S}(X,Y\otimes _{R}X)$
is a right $S$-module homomorphism given by
    $\eta _{Y}(y)(t)=y\otimes t\quad {\text{for }}t\in X.$
The counit and unit equations can now be explicitly verified. For $Y$ in ${\mathcal {D}}$,
    $\varepsilon _{FY}\circ F(\eta _{Y}):Y\otimes _{R}X\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)\otimes _{R}X\to Y\otimes _{R}X$
is given on simple tensors of $Y\otimes X$ by
    $\varepsilon _{FY}\circ F(\eta _{Y})(y\otimes x)=\eta _{Y}(y)(x)=y\otimes x.$
Likewise,
    $G(\varepsilon _{Z})\circ \eta _{GZ}:\operatorname {Hom} _{S}(X,Z)\to \operatorname {Hom} _{S}(X,\operatorname {Hom} _{S}(X,Z)\otimes _{R}X)\to \operatorname {Hom} _{S}(X,Z).$
For $\phi$ in $\operatorname {Hom} _{S}(X,Z)$,
    $G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )$
is a right $S$-module homomorphism defined by
    $G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )(x)=\varepsilon _{Z}(\phi \otimes x)=\phi (x)$
and therefore
    $G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )=\phi .$

## 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.

## In arithmetic

We can illustrate the tensor-hom adjunction in the category of functions of finite sets. Given a set $N$, its Hom functor takes any set $A$ to the set of functions from $N$ to $A$. The isomorphism class of this set of functions is the natural number $A^{N}$. Similarly, the tensor product $-\otimes N$ takes a set $A$ to its cartesian product with $N$. Its isomorphism class is thus the natural number $AN$.
This allows us to interpret the isomorphism of hom-sets
    $\operatorname {Hom} (Y\otimes X,Z)\cong \operatorname {Hom} (Y,\operatorname {Hom} (X,Z)).$
that universally characterizes the tensor-hom adjunction, as the categorification of the remarkably basic law of exponents
    $Z^{YX}=(Z^{X})^{Y}.$