  We present a theoretical study of a system consisting of a superconducting
island and two quantum dots, a possible platform for building qubits and Cooper
pair splitters, in the regime where each dot hosts a single electron and,
hence, carries a magnetic moment. We focus on the case where the dots are
coupled to overlapping superconductor states and we study whether the spins are
ferromagnetically or antiferromagnetically aligned. We show that if the total
number of particles is even, the spins align antiferromagnetically in the
flatband limit, i.e., when the bandwidth of the superconductor is negligibly
small, but ferromagnetically if the bandwidth is finite and above some value.
If the total number of particles is odd, the alignment is ferromagnetic
independently from the bandwidth. This implies that the results of the flatband
limit are applicable only within restricted parameter regime for realistic
superconducting qubit systems and that some care is required in applying
simplified models to devices such as Cooper pair splitters.
