  Hybrid stars with a quark phase described by the Nambu$-$Jona-Lasinio model
are studied. The hadron-quark model used to determine the stellar matter
equation of state favors the appearance of quark matter: the coincidence of the
deconfinement and chiral transitions and a low vacuum constituent quark mass.
These two properties are essential to build equations of state that predict
pure quark matter in the center of neutron stars. The effect of
vector-isoscalar and vector-isovector terms is discussed, and it is shown that
the vector-isoscalar terms are necessary to describe 2$M_\odot$ hybrid stars,
and the vector-isovector terms result in larger quark cores and a smaller
deconfinement density.
