Can exponential object be defined in terms of universal construction?

Question

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.

Answer 1

Yes: take the category in which an object is a pair $(X,f)$ where $X$ is an object of $C$ and $f:X\times A\to B$, and a morphism between two such objects $(X,f)$ and $(Y,g)$ is an arrow $h:X\to Y$ such that $g\circ (h\times id_A)=f$. Your given definition then says exactly that an exponential object $(B^A,eval_{AB})$ is a terminal object in this category.