Just in case anyone else is a mere mortal and takes a moment to understand Slaman's answer here is a more spelled out version of his argument. I'll just do the capping side as it's the same argument for cupping

The construction of an r.e. minimal pair and an incompatible pair of r.e. degrees whose join is $0'$ is uniform. Therefore, there are hops $H_e$ and $H_j$ such that $H_e(X) \wedge H_j(X) = X$ (non-trivially). Now, by the theorem in the psudeojump paper we can invert a hop, i.e., we can find r.e. A such that $H_e(A)= 0'$ providing the desired $A$.

Just in case anyone else is a mere mortal and takes a moment to understand Slaman's answer here is a more spelled out version of his argument. I'll just do the capping side as it's the same argument for cupping

The construction of an r.e. minimal pair (for the other side an incompatible pair of r.e. degrees whose join is $0'$) is uniform. Therefore, there are hops $H_e$ and $H_j$ such that $H_e(X) \wedge H_j(X) = X$ (non-trivially). Now, by the theorem in the psudeojump paper we can invert a hop, i.e., we can find r.e. A such that $H_e(A)= 0'$ providing the desired $A$.