Evolution of the maximum flux νF_ν(t_growth) (at the end of the growth phase t_growth = 75 r_g/c) as a function of the magnetic field. We show the results of EMBLEM without cooling (red crosses) as in Fig. C.2. Allowing the cooling during the growth phase results in a lower maximum flux (blue crosses). The maximum flux with cooling can be estimated through Eq. C.8 (black line) which is divided in two regimes, the equilibrium regime where the magnetic field is strong enough to compensate the injection and creates a stationary state (B ≥ 16.2 G) and nonstationary regime where not all electrons has cooled at t_growth (B < 16.2 G). The relative error between the analytical formula and the results of EMBLEM (with cooling) is below 30% in the whole domain and below 7% in the nonstationary regime.