  Important correspondences in representation theory can be regarded as
restrictions of the Morita--Tachikawa correspondence. Moreover, this
correspondence motivates the study of many classes of algebras like Morita
algebras and gendo-symmetric algebras. Explicitly, the Morita--Tachikawa
correspondence describes that endomorphism algebras of generators-cogenerators
over finite-dimensional algebras are exactly the finite-dimensional algebras
with dominant dimension at least two.
  In this paper, we introduce the concepts of quasi-generators and
quasi-cogenerators which generalise generators and cogenerators, respectively.
  Using these new concepts, we present higher versions of the Morita--Tachikawa
correspondence that takes into account relative dominant dimension with respect
to a self-orthogonal module with arbitrary projective and injective dimension.
These new versions also hold over Noetherian algebras which are finitely
generated and projective over a commutative Noetherian ring.
