\begin{definition}[Definition:Exponential (Category Theory)/Transpose]
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 $g$.
\end{definition}