  By modelling the expansion of a cloud of electrons and positrons with the
temperature 400 keV that propagates at the mean speed 0.9c ($c:$ speed of
light) through an initially unmagnetized electron-proton plasma with a
particle-in-cell (PIC) simulation, we find a mechanism that collimates the pair
cloud into a jet. A filamentation instability develops between the protons at
rest and the moving positrons. Its magnetic field collimates the positrons and
drives an electrostatic shock into the electron-proton plasma. The magnetic
field acts as a discontinuity that separates the protons of the shocked ambient
plasma, known as the outer cocoon, from the jet's interior region. The outer
cocoon expands at the speed 0.15c along the jet axis and at 0.03c
perpendicularly to it. The filamentation instability converts the jet's
directed flow energy into magnetic energy in the inner cocoon. The magnetic
discontinuity cannot separate the ambient electrons from the jet electrons.
Both species rapidly mix and become indistinguishable. The spatial distribution
of the positive charge carriers is in agreement with the distributions of the
ambient material and the jet material predicted by a hydrodynamic model apart
from a dilute positronic outflow that is accelerated by the electromagnetic
field at the jet's head.
