  We consider a chiral gauge theory from which light composite Dirac neutrinos
dynamically emerge, augmented by a QCD axion to solve the strong CP problem. We
show that an interplay between the composite sector and the axion can also
simultaneously lead to successful baryogenesis and generate a correct dark
matter abundance via "co-genesis" without contradicting present constraints. We
predict $\Delta N_{\rm eff} \geq 0.14$ and $m_a$ in the range $O(1)\>{\rm \mu
eV}$--$O(10)\>{\rm meV}$, which can be firmly tested by upcoming CMB
experiments and axion searches.
