In other words,
a -plethory is a commutative k-algebra together with a comonad structure on the covariant functor it represents, much as a k-algebra is the same as a -module that represents a comonad. So, just as a -algebra is exactly the structure that knows how to act on a -module, a -plethory is the structure that knows how to act on a commutative -algebra. (BB05)
The most famous example of such an object is , the ring of symmetric polynomials in countably many variables, which is a -plethory.