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To achieve the more significant passive beamforming gain in the double-intelligent reflecting surface (IRS) aided system over the conventional single-IRS counterpart, channel state information (CSI) is indispensable in practice but also more challenging to acquire, due to the presence of not only the single- but also double-reflection links that are intricately coupled and also entail more channel coefficients for estimation. In this paper, we propose a new and efficient channel estimation scheme for the double-IRS aided multi-user multiple-input multiple-output (MIMO) communication system to resolve the cascaded CSI of both its single- and double-reflection links. First, for the single-user case, the single- and double-reflection channels are efficiently estimated at the multi-antenna base station (BS) with both the IRSs turned ON (for maximal signal reflection), by exploiting the fact that their cascaded channel coefficients are scaled versions of their superimposed lower-dimensional CSI. Then, the proposed channel estimation scheme is extended to the multi-user case, where given an arbitrary user's cascaded channel (estimated as in the single-user case), the other users' cascaded channels can also be expressed as lower-dimensional scaled versions of it and thus efficiently estimated at the BS. Simulation results verify the effectiveness of the proposed channel estimation scheme and joint training reflection design for double IRSs, as compared to various benchmark schemes.