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Abstract The classical double copy relates exact solutions in biadjoint scalar, gauge and gravity theories. Recently, nonperturbative solutions have been found in biadjoint theory, which have been speculated to be related to the Wu-Yang monopole in gauge theory. We show that this seems not to be the case, by considering monopole solutions in the infinitely boosted (shockwave) limit. Furthermore, we show that the Wu-Yang monopole is instead related to the Taub-NUT solution, whose previously noted single copy is that of an abelian- like (Dirac) monopole. Our results demonstrate how abelian and non-abelian gauge theory objects can be associated with the same gravity object, and clarify a number of open questions concerning the scope of the classical double copy.