  We present the results of a full cosmological simulation with the new code
SCALAR, where dark matter is in form of fuzzy dark matter, described by a light
scalar field with a mass of $m_{\rm B} = 2.5 \times 10^{-22}$ eV and evolving
according to the Schr\"{o}dinger-Poisson system of equations. In comoving
units, the simulation volume is $2.5 ~ h^{-1} {\rm Mpc}$ on a side, with a
resolution of $20~h^{-1}{\rm pc}$ at the finest refinement level. We analyse
the formation and the evolution of central solitonic cores, which are found to
leave their imprints on dark matter density profiles, resulting in shallower
central densities, and on rotation curves, producing an additional circular
velocity peak at small radii from the center. We find that the suppression of
structures due to the quantum nature of the scalar field results in an
shallower halo mass function in the low-mass end compared to the case of a
$\Lambda$CDM simulation, in which dark matter is expected to cluster at all
mass scales even if evolved with the same initial conditions used for fuzzy
dark matter. Furthermore, we verify the scaling relations characterising the
solution to the Schr\"{o}dinger-Poisson system, for both isolated and merging
halos, and we find that they are preserved by merging processes. We
characterise each fuzzy dark matter halo in terms of the dimensionless quantity
$\Xi \propto \left | E_{\rm halo} \right |/M_{\rm halo}^3$ and we show that the
core mass is tightly linked to the halo mass by the core-halo mass relation
$M_{\rm core}/M_{\rm halo} \propto \Xi^{1/3}$. We also show that the core
surface density of the simulated fuzzy dark matter halos does not follow the
scaling with the core radius as observed for dwarf galaxies, representing a big
challenge for the fuzzy dark matter model as the sole explanation of core
formation.
