  The S\'ersic $R^{1/n}$ model is the best approximation known to date for
describing the light distribution of stellar spheroidal and disk components,
with the S\'ersic index $n$ providing a direct measure of the central radial
concentration of stars. The S\'ersic index of a galaxy's spheroidal component,
$n_{sph}$, has been shown to tightly correlate with the mass of the central
supermassive black hole, $M_{BH}$. The $M_{BH}-n_{sph}$ correlation is also
expected from other two well known scaling relations involving the spheroid
luminosity, $L_{sph}$: the $L_{sph}-n_{sph}$ and the $M_{BH}-L_{sph}$.
Obtaining an accurate estimate of the spheroid S\'ersic index requires a
careful modelling of a galaxy's light distribution and some studies have failed
to recover a statistically significant $M_{BH}-n_{sph}$ correlation. With the
aim of re-investigating the $M_{BH}-n_{sph}$ and other black hole mass scaling
relations, we performed a detailed (i.e.~bulge, disks, bars, spiral arms,
rings, halo, nucleus, etc.) decomposition of 66 galaxies, with directly
measured black hole masses, that had been imaged at $3.6\rm~\mu m$ with
Spitzer. In this paper, the third of this series, we present an analysis of the
$L_{sph}-n_{sph}$ and $M_{BH}-n_{sph}$ diagrams. While early-type
(elliptical+lenticular) and late-type (spiral) galaxies split into two separate
relations in the $L_{sph}-n_{sph}$ and $M_{BH}-L_{sph}$ diagrams, they reunite
into a single $M_{BH} \propto n_{sph}^{3.39 \pm 0.15}$ sequence with relatively
small intrinsic scatter ($\epsilon \simeq 0.25 \rm~dex$). The black hole mass
appears to be closely related to the spheroid central concentration of stars,
which mirrors the inner gradient of the spheroid gravitational potential.
