  Context. Stellar spin-down is the result of a complex process involving
rotation, dynamo, wind and magnetism. Multi-wavelength surveys of solar-like
stars have revealed the likely existence of relationships between their
rotation, X-ray luminosity, mass-losses and magnetism. Those impose strong
constraints on the corona and wind of cool stars.
  Aims. We aim to provide power-law prescriptions of the mass-loss of stars, of
their magnetic field, and of their base coronal density and temperature that
are compatible with their observationally-constrained spin-down.
  Methods. We link the magnetic field and the mass-loss rate from a wind torque
formulation in agreement with the distribution of stellar rotation periods in
open clusters and the Skumanich law. Given a wind model and an expression of
the X-ray luminosity from radiative losses, we constrain the coronal properties
by assuming different physical scenarii linking closed loops to coronal holes.
  Results. We find that the magnetic field and the mass loss are involved in a
one-to-one correspondence constrained from spin-down considerations. We show
that a magnetic field depending on both the Rossby number and the stellar mass
is required to keep a consistent spin-down model. The estimates of the magnetic
field and the mass-loss rate obtained from our formalism are consistent with
statistical studies as well as individual observations and give new leads to
constrain the magnetic field-rotation relation. The set of scaling-laws we
derived can be broadly applied to cool stars from the PMS to the end of the MS,
and allow for a stellar wind modelling consistent with all the observational
constraints available to date.
