exponential object

{{category theory}} Given objects ''Y'' and ''Z'', the '''exponential object''' <math> Z^Y </math> is uniquely defined by the following universal property: for any object ''X'' with arrow <math> g: X \times Y \rightarrow Z</math>, there can always be constructed an arrow <math> \lambda g: X \rightarrow Z^Y </math> which induces an arrow <math> \lambda g \times \hbox{id}_Y = i </math>, &nbsp; <math> i: X \times Y \rightarrow Z^Y \times Y </math> which is unique in satisfying <math> \text{eval} \circ i = g </math> where <math> \text{eval}: Z^Y \times Y \rightarrow Z </math>.