Definition:Exponentiation Functor
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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:
Object functor: | \(\ds C^A := C^A \) | $C^A$ is the exponential of $C$ by $A$ | |||||||
Morphism functor: | \(\ds f^A := \widetilde {\paren {f \circ \epsilon} } \) | $f: B \to C$ is a morphism of $\mathbf C$ |
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
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