400}_{- 100}$]{} keV. Finally, we replaced the phenomenological cut-off power law with the Comptonisation model [[compps]{}]{} [@compps]. We obtain a good fit (${\chi^{2}/\textrm{dof}}=7/9$) and a temperature of [$100^{+ 150}_{- 40}$]{} keV. We also repeated the same analysis assuming the partial-covering absorption with ${N_{\textrm{H}}}= {6.5 \times 10^{23}}$ [cm$^{-2}$]{} and a covering factor of 0.9, consistent with [[*NuSTAR*]{}]{} data [@kamraj2017], obtaining analogous results. Then we extrapolated the best-fitting cut-off power law down to 4 keV, without re-fitting, to compare it with the [[*XMM–Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra (Fig. \[fig:extrap\_4388\]). Despite the flux variability, the XMM1, XMM2, and NUS spectral shape appears in good agreement with the extrapolation, with minor discrepancies likely due to an imperfect modelling of absorption. XMM3 instead shows a significant difference in spectral shape.

![\[fig:ratios\] Residuals of fits of IBIS spectrum with different models. Upper panel: simple power law. Second panel: power law plus reflection ([[pexrav]{}]{}). Third panel: exponentially cut-off power law. Lower panel: thermal Comptonisation model ([[compps]{}]{}). ](ratios_4388){width="\columnwidth"}

![\[fig:extrap\_4388\] Upper panel: [[*XMM–Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra of NGC 4388 with the cut-off power law that best fits IBIS. Lower panel: data/model ratio. Only [[*NuSTAR*]{}]{}’s FPMA data are shown for clarity. The data were binned for plotting purposes.](extrap_4388){width="\columnwidth"}

### The broad-band 