The authors in [Math. Nachr. 282, No. 8, 1111–1133 (2009; Zbl 1181.47076)] introduced the notion of compatibility between a norm ideal of \(n\)-homogeneous polynomials \(\mathcal U_n\) and a norm ideal of linear operators \(\mathcal U\) by assuming the existence of positive constants \(A,B\) such that, for all Banach spaces \(E\) and \(F\), the following two conditions hold: (i) for each polynomial \(P\in \mathcal U_n(E,F)\) and \(a\in E\), one has that \(P_{a^{n-1}}\in \mathcal U(E,F)\) and \(\|P_{a^{n-1}}\|_{\mathcal U(E,F)}\leq A \|P\|_{\mathcal U_n(E,F)}\|a\|^{n-1}\) and (ii) for each \(T\in {\mathcal U(E,F)}\) and \(\gamma\in E'\), one has that \(\gamma^{n-1}T\in {\mathcal U_n(E,F)} \) and \(\|\gamma^{n-1}T\|_{\mathcal U_n(E,F)}\leq B \|T\|_{\mathcal U(E,F)}\|\gamma\|^{n-1}\). The main theorem of the paper shows that, for any Banach ideal of \(n\)-homogeneous polynomials \(\mathcal U_n\), there exists a unique Banach ideal of operators \(\mathcal U\) compatible with it and, in the complex case, the constants \(1\leq A, B\leq e\). A similar result is also given for the notion of coherent sequence of ideals of \(k\)-homogeneous polynomials.