In a strict symmetric monoidal category with symmetry , a map , where is a comonoid and a monoid, is a measuring if
where we wrote etc. If is in fact a bimonoid and if the measuring is an action, then is said to be a Hopf action. In the -linear case, a -algebra equipped with a Hopf action is called also a left -module algebra; it is the same as a monoid (=algebra) in the monoidal category of left -modules, where the monoidal structure is induced by the coaction . It is straightforward to modify the condition above to the case of non-strict symmetric monoidal categories. A dual concept is a Hopf coaction or equivalently, a notion of a -comodule algebra.