From Basic Category Theory for Computer Scientists
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 is an arrow $eval_{AB} : (B^A \times A) \to B$ such that for any object $C$ and arrow $g: (C \times A)\to B$ there is a unique arrow $curry(g): C \to B^A$ such that $eval_{AB} \circ (curry(g) \times id_A) =g$.
Can exponential object be defined in terms of universal construction?
A universal construction describes a class of objects and accompanying arrows that share a common property and picks out the objects that are terminal when this class is considered as a category.