  We consider how Morita equivalences are compatible with the notion of a
corner subring. Namely, we outline a canonical way to replace a corner subring
of a given ring with one which is Morita equivalent, and look at how such an
equivalence ascends. We use the language of Morita contexts, and then specify
these more general results. We give applications to trivial extensions of
finite-dimensional algebras, tensor rings of pro-species, semilinear clannish
algebras arising from orbifolds, and functor categories.
