section_path: Powers of sets/In category theory
section_name: In category theory
paragraph_text: In category theory
  
  In the category of sets, the morphisms between sets X and Y are the functions from X to Y. It results that the set of the functions from X to Y that is denoted Y X {\displaystyle Y^{X}} in the preceding section can also be denoted hom (X, Y). {\displaystyle \hom(X,Y).} The isomorphism (S T) U ≅ S T × U {\displaystyle (S^{T})^{U}\cong S^{T\times U}} can be rewritten
  
  This means the functor "exponentiation to the power T " is a right adjoint to the functor "direct product with T ".
  
  This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor X → X T {\displaystyle X\to X^{T}} is, if it exists, a right adjoint to the functor Y → T × Y. {\displaystyle Y\to T\times Y.} A category is called a Cartesian closed category, if direct products exist, and the functor Y → X × Y {\displaystyle Y\to X\times Y} has a right adjoint for every T.