  What is an adequate extension of an operator ideal I to the polynomial and
multilinear settings? This question motivated the appearance of the concepts of
coherent sequences of polynomial ideals and compatibility of a polynomial ideal
with an operator ideal, introduced by D. Carando el al. We propose a different
approach by considering pairs (U_{k},M_{k})_{k=1}^{\infty}, where
(U_{k})_{k=1}^{\infty} is a polynomial ideal and (M_{k})_{k=1}^{\infty} is a
multi-ideal, instead of considering just polynomial ideals. Our approach ends a
discomfort caused by the previous theory: for real scalars the canonical
sequence (P_{k})_{k=1}^{\infty} of continuous k-homogeneous polynomials is not
coherent according to the definition of Carando et al.
