  Using four-wave mixing microscopy, we measure the coherent response and
ultrafast dynamics of excitons and trions in MoSe$_2$ monolayers grown by
molecular beam epitaxy on thin films of hexagonal boron nitride. We assess
inhomogeneous and homogeneous broadenings in the transition spectral lineshape.
The impact of phonons on the homogeneous dephasing is inferred via the
temperature dependence of the dephasing. Four-wave mixing mapping, combined
with the atomic force microscopy, reveals spatial correlations between exciton
oscillator strength, inhomogeneous broadening and the sample morphology. The
quality of coherent optical response of the epitaxially grown transition metal
dichalcogenides becomes now comparable with the samples produced by mechanical
exfoliation, enabling coherent nonlinear spectroscopy of innovative materials,
like magnetic layers or Janus semiconductors.
