  If two cluster-tilting objects of an acyclic cluster category are related by
a mutation, then their endomorphism algebras are nearly-Morita equivalent
[Buan-Marsh-Reiten], i.e. their module categories are equivalent "up to a
simple module". This result has been generalised by D. Yang, using a result of
P-G. Plamondon, to any simple mutation of maximal rigid objects in a
2-Calabi--Yau triangulated category. In this paper, we investigate the more
general case of any mutation of a (non-necessarily maximal) rigid object in a
triangulated category with a Serre functor. In that setup, the endomorphism
algebras might not be nearly-Morita equivalent and we obtain a weaker property
that we call pseudo-Morita equivalence. Inspired by results of Buan-Marsh, we
also describe our result in terms of localisations.
