Semiconductor quantum dots in close proximity to superconductors may provoke localized bound states within the superconducting energy gap known as Yu-Shiba-Rusinov (YSR) state, which is a promising candidate for constructing Majorana zero modes and topological qubits. Side-coupled double quantum dot systems are ideal platforms revealing the secondary proximity effect. Numerical renormalization group calculations show that, if the central quantum dot can be treated as a noninteracting resonant level, it acts as a superconducting medium due to the ordinary proximity effect. The bound state in the side dot behaves as the case of a single impurity connected to two superconducting leads. The side dot undergoes quantum phase transitions between a singlet state and a doublet state as the Coulomb repulsion, the interdot coupling strength, or the energy level sweeps. Phase diagrams indicate that the phase boundaries could be well illustrated by $\Delta \approx c {T_{K2}}$ in all cases, with $\Delta$ is the superconducting gap, $T_{K2}$ is the side Kondo temperature and $c$ is of the order $1.0$. These findings offer valuable insights into the secondary proximity effect, and show great importance for designing superconducting quantum devices.
