  Non-relativistic, magnetically-driven jets are constructed by taking
self-consistently into account the feedback on the underlying accretion disc.
It is shown that such jets are mostly described by the ejection index $\xi=
\mbox{d} \ln \dot M_a/ \mbox{d}\ln r$, which is a local measure of the disc
ejection efficiency. This parameter is found to lie in a very narrow range, due
to global disc-jets constraints. This investigation provided two important
results. First, the disc vertical equilibrium imposes a minimum mass flux
ejected. Second, their asymptotic behaviour critically depends on a fastness
parameter $\omega_A= \Omega_*r_A /V_{Ap,A}$, ratio of the field lines rotation
velocity to the poloidal Alfv\'en velocity at the Alfv\'en surface. This
parameter must be bigger than, but of the order of, unity. Self-similar jets
from Keplerian discs, after widening up to a maximum radius whose value
increases with $\omega_A$, always recollimate towards the jet axis, until the
fast-magnetosonic critical point is reached. It is doubtful that such solutions
could steadily cross this last point, the jet either ending there or
rebouncing. Recollimation takes place because of the increasing effect of
magnetic constriction. This systematic behaviour is due to the large opening of
the magnetic surfaces, leading to such an efficient acceleration that matter
always reaches its maximum poloidal velocity. This ``over-widening'' stems from
having the same ejection efficiency $\xi$ in the whole jet. Realistic jets, fed
with ejection indices varying from one magnetic surface to the other, would not
undergo recollimation, allowing either cylindrical or parabolic asymptotic
collimation.
