  We investigate the effects of electron-phonon coupling, as well as of spin
and charge fluctuations on the superconducting state in a single layer of FeSe
on SrTiO$_3$ substrate. These three bosonic mediators of Cooper pairing are
treated on equal footing in a multichannel, full-bandwidth, multiband, and
anisotropic Eliashberg theory of the interacting state. Our self-consistent
calculations show that an s-wave symmetry of the superconducting gap is
compatible only with a complete absence of spin fluctuations. When spin
fluctuations are present, the sign-changing nodeless d-wave pairing symmetry is
always obtained, yet the essential ingredient for explaining the gap magnitude
and critical temperature is still the interfacial electron-phonon interaction.
