Definition:Exponentiation Functor

Theorem
Let $\mathbf C$ be a Cartesian closed metacategory.

Let $A$ be an object of $\mathbf C$.

Then exponentiation by $A$, denoted $\paren -^A: \mathbf C \to \mathbf C$, is the functor defined by:

Here $\epsilon: B^A \times A \to B$ denotes the evaluation morphism, and $\widetilde {\paren {f \circ \epsilon} }: B^A \to C^A$ is the exponential transpose of $f \circ \epsilon$.

That it is in fact a functor is shown on Exponentiation Functor is Functor.

Also see

 * Definition:Exponential (Category Theory)