  We study the critical Josephson current flowing through a double quantum dot
weakly coupled to two superconducting leads. We use analytical as well as
numerical methods to investigate this setup in the limit of small and large
bandwidth leads in all possible charging states, where we account for on-site
interactions exactly. Our results provide clear signatures of nonlocal
spin-entangled pairs, which support interpretations of recent experiments
[Deacon, R. S. et al., Nat. Commun. 6, 7446 (2015)]. In addition, we find that
the ground state with one electron on each quantum dot can undergo a tunable
singlet-triplet phase transition in the regime where the superconducting gap in
the leads is not too large, which gives rise to an additional new signature of
nonlocal Cooper pair transport.
