Various engine types are thermodynamically equivalent in the quantum limit of small “engine action”. Our previous derivation of the equivalence is restricted to Markovian heat baths and to implicit classical work repository (e.g., laser light in the semi-classical approximation). In this paper, all the components, baths, batteries, and engines, are explicitly taken into account. To neatly treat non-Markovian dynamics, we use mediating particles that function as a heat exchanger. We find that, on top of the previously observed equivalence, there is a higher degree of equivalence that cannot be achieved in the Markovian regime. Next, we focus on the quality of the battery charging process. A condition for positive energy increase and zero entropy increase (work) is given. Moreover, it is shown that, in the strong coupling regime, it is possible to super-charge a battery. With super-charging, the energy of the battery is increased while its entropy is being reduced at the same time.