  Relativistic electrons are an essential component in many astrophysical
sources, and their radiation may dominate the high-energy bands. Inverse
Compton (IC) emission is the radiation mechanism that plays the most important
role in these bands. The basic properties of IC, such as the total and
differential cross sections, have long been studied; the properties of the IC
emission depend strongly not only on the emitting electron distribution but
also on the properties of the target photons. This complicates the
phenomenological studies of sources, where target photons are supplied from a
broad radiation component. We study the spectral properties of IC emission
generated by a power-law distribution of electrons on a power-law distribution
of target photons. We approximate the resulting spectrum by a broken-power-law
distribution and show that there can be up to three physically motivated
spectral breaks. If the target photon spectrum extends to sufficiently low
energies, $\varepsilon_{\mathrm{min}}< m_e^2c^4/E_{\mathrm{max}}$ ($m_e$ and
$c$ are electron mass and speed of light, respectively;
$\varepsilon_{\mathrm{min}}$ and $E_{\mathrm{max}}$ are the minimum/maximum
energies of target photons and electrons, respectively), then the high energy
part of the IC component has a spectral slope typical for the Thomson regime
with an abrupt cutoff close to $E_{\mathrm{max}}$. The spectra typical for the
Klein-Nishina regime are formed above $m_e^2c^4/\varepsilon_{\mathrm{min}}$. If
the spectrum of target photons features a cooling break, i.e., a change of the
photon index by $0.5$ at $\varepsilon_{\mathrm{br}}$, then the transition to
the Klein-Nishina regime proceeds through an intermediate change of the photon
index by $0.5$ at $m_e^2c^4/\varepsilon_{\mathrm{br}}$.
