  We employ a set of Magneticum cosmological hydrodynamic simulations that span
over $15$ different cosmologies, and extract masses and concentrations of all
well-resolved haloes between $z=0-1$ for critical over-densities
$\Delta_\texttt{vir}, \Delta_{200c}, \Delta_{500c}, \Delta_{2500c}$ and mean
overdensity $\Delta_{200m}.$ We provide the first mass-concentration (Mc)
relation and sparsity relation (i.e. $M_{\Delta1} - M_{\Delta2}$ mass
conversion) of hydrodynamic simulations that is modelled by mass, redshift and
cosmological parameters $\Omega_m, \Omega_b, \sigma_8, h_0$ as a tool for
observational studies. We also quantify the impact that the Mc relation scatter
and the assumption of NFW density profiles have on the uncertainty of the
sparsity relation. We find that converting masses with the aid of a Mc relation
carries an additional fractional scatter ($\approx 4\%$) originated from
deviations from the assumed NFW density profile. For this reason we provide a
direct mass-mass conversion relation fit that depends on redshift and
cosmological parameters. We release the package hydro\_mc, a python tool that
perform all kind of conversions presented in this paper.
