I'm currently introducing myself to the ring of Witt vectors. One comment that comes up quite often, is that the ring of Witt vectors can be thought of as an arithmetic analogues of the ring of formal power series, or that they have a similar role (for instance https://ncatlab.org/nlab/show/ring+of+Witt+vectors).
I know that the ring of p-typical Witt vectors are right adjoint functors, since there is the "Taylor Series functor" $$Hom_{CRing}(R,S) \cong Hom_{p-derivation}((R,\delta_p), W(S))$$ (see for instance page 5 of these notes of a lecture given by J.Borger https://maths-people.anu.edu.au/~borger/classes/copenhagen-2016/LectureNotes.pdf) This is similar to the bijection $$ Hom_{CRing}(R,S) \cong Hom_{\delta}((R,\delta),(S[[t]],\frac{d}{dt}))$$ which is used in differential algebra.
However, for the ring of formal power series we know that $R[[t]]$ has the universal property that if $S$ is a ring, and $x\in S$ is in an ideal $I\subset S$ such that $S$ is complete with regard to the $I$-adic topology, then there exits a unique continuous ring homomorphism $R[[t]] \rightarrow S$ such that $t\mapsto x$. Thus the functor which associates to a ring $R$ the formal power series ring $R[[t]]$ is free in the category of complete rings.
Lead by this analogy, I would like to know if we have a similar bijection $Hom(W(R),S) \cong \ldots$ in the correct category?