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.
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