  Ni-Mn-In magnetic shape-memory Heusler alloys exhibit generally a large
thermal hysteresis at their first-order martensitic phase transition which
hinder a technological application in magnetic refrigeration. By optimizing the
Cu content in Ni$_2$Cu$_x$Mn$_{1.4-x}$In$_{0.6}$, we obtained a thermal
hysteresis of the martensitic phase transition in
Ni$_{2}$Cu$_{0.2}$Mn$_{1.2}$In$_{0.6}$ of only 6 K. We can explain this very
small hysteresis by an almost perfect habit plane at the interface of
martensite and austenite phases. Application of hydrostatic pressure does not
reduce the hysteresis further, but shifts the martensitic transition close to
room temperature. The isothermal entropy change does not depend on warming or
cooling protocols and is pressure independent. Experiments in pulsed-magnetic
fields on Ni$_{2}$Cu$_{0.2}$Mn$_{1.2}$In$_{0.6}$ find a reversible
magnetocaloric effect with a maximum adiabatic temperature change of -13 K.
