  The collimation of a Poynting-flux dominated jet by a wind emanating from the
surface of an accretion flow is computed using a semi-analytic model. The
injection of the disk wind is treated as a boundary condition in the equatorial
plane, and its evolution is followed by invoking a prescribed geometry of
streamlines. Solutions are obtained for a wide range of disk wind parameters.
It is found that jet collimation generally occurs when the total wind power
exceeds about ten percents of the jet power. For moderate wind powers we find
gradual collimation. For strong winds we find rapid collimation followed by
focusing of the jet, after which it remains narrow over many Alfv\'en crossing
times before becoming conical. We estimate that in the later case the jet's
magnetic field may be dissipated by the current-driven kink instability over a
distance of a few hundreds gravitational radii. We apply the model to M87 and
show that the observed parabolic shape of the radio jet within the Bondi radius
can be reproduced provided that the wind injection zone extends to several
hundreds gravitational radii, and that its total power is about one third of
the jet power. The radio spectrum can be produced by synchrotron radiation of
relativistically hot, thermal electrons in the sheath flow surrounding the
inner jet.
