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tokens: to T bb > 10 5 K, n H 10 12 cm −3 or ξ 0 10 4 . In this region of the parameter space, the properties of the warm corona are mainly driven by χ . However, for smaller T bb or larger density, n H , or ionization parameter, ξ 0 , the radiative and spectral properties of the corona depend significantly on these parameter values. The background colors in Fig. 5 correspond to the Compton-cooling-to-corona-heating ratio p compt / q h , whose color scale is reported on the right of the figure. The dependencies of the corona emission spectral shape on T bb and n H can be explained by the properties of the Comptonization process in an environment in radiative equilibrium. At low density or high disk temperature, our simulations show that the corona emission is dominated by Comptonization. In this case, as long as the heating over cooling of the warm corona remains constant, no significant spectral variability is expected. The corona heating is directly related to the sum of the internal heating Q and the illumination ξ 0 while the cooling is related to . They are all related to one another through the net flux crossing the base of the corona (see Eq. ( 10 )). As long as the illumination is negligible with respect to warm corona internal heating power, a constant χ implies a constant heating-to-cooling ratio and the warm corona temperature is not expected to change. The warm corona optical depth being fixed, the spectral shape of the warm corona emission remains roughly constant. This is very similar to the constancy of the spectral shape of the emission from a hot optically thin corona in radiative equilibrium with the surrounding soft photon field (e.g., Haardt Maraschi 1991 ; Haardt 1993 ; Petrucci et al. 2000 ; Tortosa et al. 2018 ). However, for densities larger than ∼10 12 cm −3 and disk temperatures lower than ∼10 5 K, the dependence on these parameters becomes important. This is due to the fact that in this part of the parameter space, the corona radiative processes are less dominated by Comptonization, either because the soft seed photon flux decreases (at low T bb ) or the free-free and bound-free processes increase (at high density) through their dependence on the density squared. The previous discussion in terms of radiative equilibrium no longer holds. At constant χ , decreasing T bb or increasing n H results in a decrease of the temperature and a steepening of the warm corona spectrum. These conditions are thus less favorable to producing spectra that could reach the soft X-ray band and produce a significant soft X-ray excess. The dependence of the spectral shape and temperature of the warm corona on the ionization parameter ξ 0 is quite different. The spectral shape and temperature are almost independent of ξ 0 unless this latter becomes larger than a few 10 4 erg s −1 cm. While for such values of ξ 0 , the warm corona is dominated by Compton cooling, the spectral shape (photon index and temperature) varies with ξ 0 and the lower the χ