Playing Civ V with Steam on different PC

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My friend and I want to play Civ V together using LAN. His laptop isn't te newest one and has some performance issues so we thought that he could play on mine using his Steam account since I have two computers anyway.
Would it be a problem since we haven't purchased the same DLCs? Would Steam use the version of Civilisation that I have installed on my PC or what he has bought and linked to his account?
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Sources: Stack Exchange - This article follows the attribution requirements of Stack Exchange and is licensed under CC BY-SA 3.0.
Images: Tima Miroshnichenko, Alexander Kovalev, Yan Krukov, Yan Krukov