The usual decategorification $K(C)$ of a monoidal category $C$ is by taking the isomorphic classes of $C$ as objects. The resulting decategorification $K(C)$ is a monoid.
There is another decategorification that takes the morphisms as objects instead. The result is also a monoid. Is there a name for and literature on this?
EDIT: Let me clarify after Boris' comment. Start with a pointed monoidal category $C$. Then the set of arrows $Arr(C)$ is a monoid with the tensor product and the identity arrow.