Definition:Exponential (Category Theory)/Transpose

Definition
Let $\mathbf C$ be a metacategory with binary products.

Let $B$ and $C$ be objects of $\mathbf C$.

Let $C^B$ be an exponential of $C$ by $B$, with evaluation morphism $\epsilon: C^B \times B \to C$.

For a morphism $f: A \times B \to C$, the unique:


 * $\tilde f: A \to C^B$

provided by the UMP for $C^B$ is called the exponential transpose of $f$.

For a morphism $g: A \to C^B$, the morphism $\bar g: A \times B \to C$ defined by:


 * $\bar g = \epsilon \circ \paren {g \times \operatorname{id}_B}$

is also called the exponential transpose of $g$.

Also see

 * Exponential Transpose of Exponential Transpose