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Semiconductor InSb nanosheet/hexagonal boron nitride (hBN)/graphite trilayers are fabricated, and single- and double-gate devices made from the trilayers are realized and characterized. The InSb nanosheets employed in the trilayer devices are epitaxially grown, free-standing, zincblende crystals and are in micrometer lateral sizes. The hBN and graphite flakes are obtained by exfoliation. Each trilayer is made by successively stacking an InSb nanosheet on an hBN flake and on a graphite flake using a home-made alignment stacking/transfer setup. The fabricated single- and double-gate devices are characterized by electrical and/or magnetotransport measurements. In all these devices, the graphite and hBN flakes are employed as the bottom gates and the gate dielectrics. The measurements of a fabricated single bottom-gate field-effect device show that the InSb nanosheet in the device has an electron field-effect mobility of ~7300 cm$^2$V$^{-1}$s$^{-1}$ and a low gate hysteresis of ~0.05 V at 1.9 K. The measurements of a double-gate Hall-bar device show that both the top and the bottom gate exhibit strong capacitive couplings to the InSb nanosheet channel and can thus tune the nanosheet channel conduction effectively. The electron Hall mobility in the InSb nanosheet of the Hall-bar device is extracted to be larger than 1.1$\times$10$^4$ cm$^2$V$^{-1}$s$^{-1}$ at a sheet electron density of ~6.1$\times$10$^{11}$ cm$^{-2}$ and 1.9 K and, thus, the device exhibits well-defined Shubnikov-de Haas oscillations.