A right adjoint to the truncated Witt functor?

For any ring $A$, let $\mathrm{wEt}_A$ be the category of weakly etale $A$-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor
$$W_r : \mathrm{wEt}_A \longrightarrow \mathrm{wEt}_{W_r(A)}$$
is well-defined and commutes with all colimits.

Can one explicit a right adjoint to $W_r$ ?

Answer

Attribution
Source : Link , Question Author : js21 , Answer Author : Community