Background:
In preconditioned Krylov subspace (KSP) methods (e.g. PCG, PGMRES etc.), a matrix $\boldsymbol{M \approx A}$ called preconditioner is required so that a new set of linear systems, e.g. $\boldsymbol{M^{-1} Ax = M^{-1}b}$, is much easier to solve than the original systems $\boldsymbol{Ax = b}$ (suppose it is ill-posed) since $\boldsymbol{M^{-1} A}$ has a much smaller condition number.
In many literatures the authors often point out that, multigrid (e.g. geometric multigrid) is important since it is not only one efficient type of solver alone to deal with a large set of linear systems but can also serve as a "good preconditioner" for the KSP method.
To my knowledge, although a solution $\boldsymbol{x}$ can be obtained iteratively with multigrid (e.g. via doses of V-cycles) like any other iterative methods, it seems that a precondition matrix $\boldsymbol{M}$ needed by the KSP methods is not generated during the entire procedure of the multigrid method.
Questions:
What does the so-called "multigrid preconditioner" exactly means?
In other words, how can a multigrid solver function as a preconditioner?
The combination of smoothing / low-pass filtering $\bf L$ and subsampling $\bf S$ is an approximation so that $$\bf (SL)^{-1}SLA \approx A$$ Now $({\bf SL})^{-1}$ may not actually exist, but we can approximate it or use regularization to avoid small eigenvalues wreaking havoc.
What a multigrid does is to solve or approximate $$\bf SLAx = SLb$$ for different $\bf S,L$.
Do you see where $\bf M$ comes into the picture?