  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 engine, 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 energy transfer to the work repository. A condition for
zero entropy increase in the work repository is given. Moreover, it is shown
that in the strong coupling regime it is possible to charge a battery with
energy while reducing its entropy at the same time.
