  In this fourth paper from the AGORA Collaboration, we study the evolution
down to redshift $z=2$ and below of a set of cosmological zoom-in simulations
of a Milky Way mass galaxy by eight of the leading hydrodynamic simulation
codes. We also compare this CosmoRun suite of simulations with dark matter-only
simulations by the same eight codes. We analyze general properties of the halo
and galaxy at $z=4$ and 3, and before the last major merger, focusing on the
formation of well-defined rotationally-supported disks, the mass-metallicity
relation, the specific star formation rate, the gas metallicity gradients, and
the non-axisymmetric structures in the stellar disks. Codes generally converge
well to the stellar-to-halo mass ratios predicted by semi-analytic models at
$z\sim$2. We see that almost all the hydro codes develop rotationally-supported
structures at low redshifts. Most agree within 0.5 dex with the observed MZR at
high and intermediate redshifts, and reproduce the gas metallicity gradients
obtained from analytical models and low-redshift observations. We confirm that
the inter-code differences in the halo assembly history reported in the first
paper of the collaboration also exist in CosmoRun, making the code-to-code
comparison more difficult. We show that such differences are mainly due to
variations in code-dependent parameters that control the time-stepping strategy
of the gravity solver. We find that variations in the early stellar feedback
can also result in differences in the timing of the low-redshift mergers. All
the simulation data down to $z=2$ and the auxiliary data will be made publicly
available.
