Editing Exponentiation (section)

===In category theory===
{{Main|Cartesian closed category}}
In the [[category of sets]], the [[morphism]]s between sets {{mvar|X}} and {{mvar|Y}} are the functions from {{mvar|X}} to {{mvar|Y}}. It results that the set of the functions from {{mvar|X}} to {{mvar|Y}} that is denoted <math>Y^X</math> in the preceding section can also be denoted <math>\hom(X,Y).</math> The isomorphism <math>(S^T)^U\cong S^{T\times U}</math> can be rewritten
:<math>\hom(U,S^T)\cong \hom(T\times U,S).</math>
This means the functor "exponentiation to the power {{mvar|T{{space|thin}}}}" is a [[right adjoint]] to the functor "direct product with {{mvar|T{{space|thin}}}}".

This generalizes to the definition of [[exponential (category theory)|exponentiation in a category]] in which finite [[direct product]]s exist: in such a category, the functor <math>X\to X^T</math> is, if it exists, a right adjoint to the functor <math>Y\to T\times Y.</math> A category is called a ''Cartesian closed category'', if direct products exist, and the functor <math>Y\to X\times Y</math> has a right adjoint for every {{mvar|T}}.