# Can exponential object be defined in terms of universal construction?

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.

• What do you mean ? This is a definition by a universal property – Max Jul 19 '19 at 21:21

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.