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Let $P \subseteq \{0,1\}^m$ be a non-empty predicate and let $\mu$ by a full-support distribution on $\mu$. A generalized polymorphism of $P$ is a tuple of functions $f_1,\dots,f_m\colon \{0,1\}^n \to \{0,1\}$ such that if $(x^{(1)}_i,\dots,x^{(m)}_i) \in P$ for all $i$ then also $(f_1(x^{(1)}), \dots, f_m(x^{(m)})) \in P$. The tuple $f_1,\dots,f_m$ is a $(\mu,\epsilon)$-approximate generalized polymorphism of $P$ if this property holds with probability at least $1-\epsilon$ where every $(x^{(1)}_i,\dots,x^{(m)}_i)$ is sampled according to $\mu$.

We prove that every approximate generalized polymorphism of $P$ is close to an exact generalized polymorphism of $P$. This generalizes Mossel’s quantitative Arrow’s theorem, as well our own previous work on functional predicates.