  We report on a novel exchange mechanism, mediating the Kondo screening, in a
correlated double quantum dot structure in the proximity of superconductor,
stemming from nonlocal Andreev reflection processes. We estimate its strength
perturbatively and corroborate analytical predictions with accurate numerical
renormalization group calculations using an effective model for the
superconductor-proximized nanostructure. We demonstrate that
superconductor-induced exchange leads to a two-stage Kondo screening. We
determine the dependence of the Kondo temperature on the coupling to
superconductor and predict a characteristic modification of conventional
low-temperature transport behavior, which can be used to experimentally
distinguish this phenomenon from other Kondo effects.
