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We have studied magnetocaloric and elastocaloric properties of a Ni-Mn-Sn(Cu) metamagnetic shape-memory alloy undergoing a magneto-structural transition (martensitic type) close to room temperature. Changes of entropy have been induced by isothermally applying both mechanical (uniaxial stress) and magnetic fields. These entropy changes have been, respectively, estimated from dilatometric measurements giving the length of the sample as a function of temperature at selected applied forces and magnetic fields and from magnetization measurements as a function of temperature at selected applied magnetic fields. Our results indicate that the elastocaloric effect is conventional and occurs in two steps which reflect the interplay between the martensitic and the incipient magnetic transitions. By contrast, the magnetocaloric effect is inverse and occurs in a single step that encompasses the effect arising from both transitions.