To improve our understanding of high-z galaxies, we study the impact of H-2 chemistry on their evolution, morphology and observed properties. We compare two zoom-in high-resolution (30 pc) simulations of prototypical M-star similar to 1010M(circle dot) galaxies at z = 6. The first, 'Dahlia', adopts an equilibrium model for H-2 formation, while the second, 'Alth ae a', features an improved non-equilibrium chemistry network. The star formation rate (SFR) of the two galaxies is similar (within 50 per cent), and increases with time reaching values close to 100M(circle dot) yr(-1) at z = 6. They both have SFR-stellar mass relation consistent with observations, and a specific SFR of similar or equal to 5Gyr(-1). The main differences arise in the gas properties. The non-equilibrium chemistry determines the H -> H-2 transition to occur at densities > 300 cm(-3), i.e. about 10 times larger than predicted by the equilibrium model used for Dahlia. As a result, Alth ae a features a more clumpy and fragmented morphology, in turn makingSNfeedback more effective. Also, because of the lower density and weaker feedback, Dahlia sits 3 sigma away from the Schmidt-Kennicutt relation; Alth ae a, instead nicely agrees with observations. The different gas properties result in widely different observables. Alth ae a outshines Dahlia by a factor of 7 (15) in [C (II)] 157.74 mu m (H(2)17.03 mu m) line emission. Yet, Alth ae a is underluminous with respect to the locally observed [C (II)]-SFR relation. Whether this relation does not apply at high-z or the line luminosity is reduced by cosmic microwave background and metallicity effects remain as an open question.