To study thermodynamic constraints associated with finite-sized reservoirs in more general cases, a thermodynamic cycle between two reservoirs with finite and non-symmetrical heat capacities is proposed, assuming that heat capacities satisfy Debye law
C\propto T^n 
． Here, asymmetry of reservoirs is reflected in different power exponents
n 
of heat capacity
C 
with respect to temperature
T 
of different reservoirs． For common solid materials,
n is determined by spatial dimensionality of the material and temperature range it is in． In two limit regimes of
\xi 
(defined as initial heat capacity ratio of low-temperature reservoir to high-temperature reservoir), i.e.,
\xi \to 0 
and
\xi \to \infty 
, it is proved that efficiency at maximum power (EMP) as a function of efficiency at maximum work (EMW) has a universality that is independent of dimension of the reservoir． In addition, a special case where initial heat capacities of two heat reservoirs are symmetric in numerical value but their heat capacity behaviors (with respect to temperature) are not symmetric is numerically studied． EMW and EMP are found to increase with increased dimension of high-temperature reservoir, and their monotonic behavior with changed dimension of low-temperature reservoir depends on initial temperature ratio
T_\mathrmL/T_\mathrmH 
of low- and high-temperature reservoirs． This finding suggests that if heat capacity properties of finite-sized reservoirs are incorporated into cycle optimization, optimization direction at low-temperature end (the choice of
n in a given interval) should be determined by the initial Carnot efficiency
\eta _\mathrmC\equiv 1-T_\mathrmL/T_\mathrmH 
．
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