  The primary X-ray emission in active galactic nuclei (AGNs) is widely
believed to be due to Comptonisation of the thermal radiation from the
accretion disc in a corona of hot electrons. The resulting spectra can, in
first approximation, be modelled with a cut-off power law, the photon index and
the high-energy roll-over encoding information on the physical properties of
the X-ray-emitting region. The photon index and the high-energy curvature of
AGNs ($\Gamma$, E$_c$) have been largely studied since the launch of X-ray
satellites operating above 10 keV. However, high-precision measurements of
these two observables have only been obtained in recent years thanks to the
unprecedented sensitivity of NuSTAR up to 79 keV. We aim at deriving relations
between phenomenological parameters ($\Gamma$ and E$_c$) and the intrinsic
properties of the X-ray-emitting region (the hot corona), namely the optical
depth and temperature. We use MoCA (Monte Carlo code for Comptonisation in
Astrophysics) to produce synthetic spectra for the case of an AGN with
M$_{BH}$=1.5$\times$10$^8$ M$_{sun}$ and accretion rate of 10% and then
compared them with the widely used power-law model with an exponential
high-energy cutoff. We provide phenomenological relations relating $\Gamma$ and
E$_c$ with the opacity and temperature of the coronal electrons for the case of
spherical and slab-like coronae. These relations give origin to a well defined
parameter space which fully contains the observed values. Exploiting the
increasing number of high-energy cut-offs quoted in the literature, we report
on the comparison of physical quantities obtained using MoCA with those
estimated using commonly adopted spectral Comptonisation models. Finally, we
discuss the negligible impact of different black hole masses and accretion
rates on the inferred relations.
