\begin{definition}[Definition:Exponential (Category Theory)]
Let $\mathbf C$ be a metacategory with binary products.
Let $B$ and $C$ be objects of $\mathbf C$.
An '''exponential of $C$ by $B$''' consists of an object $C^B$ of $\mathbf C$ and a morphism:
:$\epsilon: C^B \times B \to C$
subject to the following UMP:
:For all objects $A$ and morphisms $f: A \times B \to C$, there exists a unique morphism:
::$\tilde f: A \to C^B$
:such that:
:::$\begin{xy}\xymatrix@+1em@L+3px{
 C^B \times B
  \ar[r]^*+{\epsilon}
&
 C
\\
 A \times B
  \ar[u]_*+{\tilde f \times \operatorname{id}_B \hskip{2cm}}
  \ar[ur]_*+{f}
}\end{xy}$
:is a commutative diagram, i.e. $\epsilon \circ \paren {\tilde f \times \operatorname{id}_B} = f$.
\end{definition}