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 jets 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 jets 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 jets head.
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