  We generalize the classical notion of adjoint of a linear operator and the
Aron-Schottenloher notion of adjoint of a homogeneous polynomial. The general
notion is shown to enjoy several properties enjoyed by the classical ones,
nevertheless differences between the two theories are detected. The proofs of
both positive and negative results are not simple adaptations of the linear
cases, actually nonlinear arguments are often required. Applications of the
generalized adjoints to Lindstr\"om-Schl\"uchtermann type theorems for
composition operators are provided.
