Heat $Q_h$ is taken from a hot reservoir at a temperature of $T_h$, parts of it is converted to work $W$ by the heat engine, and the residual heat $Q_c$ is dumped into a cold reservoir at a lower temperature $T_c$.

The second law of thermodynamics implies that perfect refrigerators and perfect heat engines do not exist

Violating the refrigerator statement of second law violates the engine statement.