Exponential object

Let C be a category with all binary products and let $A$ and $B$ be objects of C. An object $B^{A}$ is an exponential object if there's an arrow $eval_{AB}:(B^{A}\times A)\rightarrow B$ such that for any object $C$ and arrow $g:(C\times A)\rightarrow B$ there is a unique arrow $curry(g):C\rightarrow B^{A}$ such that $eval_{AB}\times (curry(g)\times id_{A})=g$