Let $X$ and $Y$ be objects of a category$C$ such that all binary coproducts with $Y$ exist. (Usually, $C$ actually has all binary coproducts.) Then an coexponential object is an object $Coexp(Y, X)$ equipped with a coevaluation map $\eta \colon X \to Coexp(Y, X) \coprod Y$ which is universal in the sense that, given any object $Z$ and map $e\colon X \to Z \coprod Y$, there exists a unique map $u\colon Coexp(Y, X) \to Z$ such that $e = (u, id_Y) \circ \eta$.