  In this study, we investigate the boundedness of composition operators acting
on Morrey spaces and weak Morrey spaces. The primary aim of this study is to
investigate a necessary and sufficient condition on the boundedness of the
composition operator induced by a diffeomorphism on Morrey spaces. In
particular, detailed information is derived from the boundedness, i.e.,
  the bi-Lipschitz continuity of the mapping that induces the composition
operator follows from the continuity of the composition mapping. The idea of
the proof is to determine the Morrey norm of the characteristic functions, and
employ a specific function composed of a characteristic function. As the
specific function belongs to Morrey spaces but not to Lebesgue spaces,
  the result reveals a new phenomenon
  not observed in Lebesgue spaces. Subsequently, we prove the boundedness of
the composition operator induced by a mapping that satisfies a suitable volume
estimate on general weak-type spaces generated by normed spaces. As a
corollary, a necessary and sufficient condition for the boundedness of the
composition operator on weak Morrey spaces is provided.
