  Neutron stars that are cold enough should have two or more
superfluids/supercondutors in their inner crusts and cores. The implication of
superfluidity/superconductivity for equilibrium and dynamical neutron star
states is that each individual particle species that forms a condensate must
have its own, independent number density current and equation of motion that
determines that current. An important consequence of the quasiparticle nature
of each condensate is the so-called entrainment effect, i.e. the momentum of a
condensate is a linear combination of its own current and those of the other
condensates. We present here the first fully relativistic modelling of slowly
rotating superfluid neutron stars with entrainment that is accurate to the
second-order in the rotation rates. The stars consist of superfluid neutrons,
superconducting protons, and a highly degenerate, relativistic gas of
electrons. We use a relativistic $\sigma$ - $\omega$ mean field model for the
equation of state of the matter and the entrainment. We determine the effect of
a relative rotation between the neutrons and protons on a star's total mass,
shape, and Kepler, mass-shedding limit.
