Clones from comonoids

Abstract

The fact that the cocommutative comonoids in a symmetric monoidal category form the best possible approximation by a cartesian category is revisited when the original category is only braided monoidal. This leads to the question when the endomorphism operad of a comonoid is a clone (a Lawvere theory). By giving an explicit example, we prove that this does not imply that the comonoid is cocommutative.