  We provide coincidence results for vector-valued ideals of multilinear
operators. More precisely, if $\mathfrak A$ is an ideal of $n$-linear mappings
we give conditions for which the following equality $\mathfrak
A(E_1,\dots,E_n;F) = {\mathfrak A}^{min}(E_1,\dots,E_n;F)$ holds isometrically.
As an application, we obtain in many cases that the monomials form a Schauder
basis on the space $\mathfrak A(E_1,\dots,E_n;F)$. Several structural and
geometric properties are also derived using this equality. We apply our results
to the particular case where $\mathfrak A$ is the classical ideal of extendible
or Pietsch-integral multilinear operators. Similar statements are given for
ideals of vector-valued homogeneous polynomials.
