  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
$M_{\star}\sim 10^{10} {\rm M}_{\odot}$ 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\%), and
increases with time reaching values close to 100 ${\rm M}_{\odot}/\rm yr$ at
$z=6$. They both have SFR-stellar mass relation consistent with observations,
and a specific SFR of $\simeq 5\, {\rm Gyr}^{-1}$. The main differences arise
in the gas properties. The non-equilibrium chemistry determines the
H$\rightarrow$ 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 making SN feedback 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
[CII]~$157.74\,\mu{\rm m}$ (H$_{2}$~$17.03\,\mu{\rm m}$) line emission. Yet,
Alth{\ae}a is under-luminous with respect to the locally observed [CII]-SFR
relation. Whether this relation does not apply at high-z or the line luminosity
is reduced by CMB and metallicity effects remains as an open question.
