exponential object

Let \(\mathcal{C}\) be a category, \(X,Y \in \mathrm{Ob}(\mathcal{C})\) objects, such that \(\mathcal{C}\) admits products of the form \(- \times Y\). Then the exponential object, if it exists, is defined as representing object of the functor

\begin{align*} X^Y \coloneqq \mathrm{Hom}_{\mathcal{C}}(- \times Y, X) \end{align*}

see:

• right adjoint functor and objectwise representable