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Seems to be in section 5 of those lectures?
That approach doesn’t seem especially elegant, though. It is enough just to specify what the subdivision functor does to $\square_{\leq 1}$, which can be done very explicitly. One gets everything else (including a right adjoint) from the universal properties. I think I described this once in some ancient nForum posts :-). Edit: think I found the nForum posts, see from #12 here.
So I changed the relevant sentence to a more proper citation:
A cubical subdivision functor $sd$ is discussed in Jardine 02, Section 5.
Ah yes – I think when I initially posted here the link was to a different paper of Jardine’s (there are two of them in the references). I eventually tracked it down and fixed the reference, but forgot to mention it. Sorry for the wild goose chase!
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