  We calculate the Compton scattering for photon and gluon with the
Klein-Nishina formula in fixed-target collisions by using the proton and lead
beams at AFTER@LHC. In these collisions, we can investigate the particular case
of Compton scattering at the partonic level, such as $\gamma q\rightarrow
q\gamma$, $\gamma q\rightarrow qg$, $gq\rightarrow q\gamma$, and $gq\rightarrow
qg$, that can help to check of the equivalent-photon approximation and
understand the dynamics of hadron collisions at high energies, as well as probe
the inner hadron structure.
