Let $\Lambda$ be the ring of symmetric functions in infinitely many variables. There is a biring structure on $\Lambda$ defined by the coaddition $$\Delta^+: \Lambda \to \Lambda \otimes \Lambda$$ $$f(x_1, x_2, \ldots) \mapsto f(x_1,y_1, x_2, y_2, \ldots)$$ and the comultiplication $$\Delta^\times: \Lambda \to \Lambda \otimes \Lambda$$ $$f(x_1, x_2, \ldots) \mapsto f(\ldots, x_iy_j, \ldots)$$ (where we identify $\Lambda \otimes \Lambda$ with functions of two infinite sets of variables $(x_i)$ and $(y_i)$ which are symmetric in both the $x$'s and the $y$'s). This defines a lift of the hom-functor $\text{Hom}(\Lambda, -)$ to a functor $\text{CRing}\to \text{CRing}$; this is the "ring of Witt vectors" functor. Moreover, the operation of plethysm on $\Lambda$ defines a comonad structure on this functor. Coalgebras for this comonad are Grothendieck's $\lambda$-rings.