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If $F:C \to D$ is a Cartesian fibration of $\infty$-categories, I would like to show that $F$ preserves pullbacks. This seems intuitively clear, but I haven't found it in HTT (but perhaps I missed it).

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This is false already for ordinary fibrations of 1-categories. It is a theorem that if $D$ has pullbacks and $F:C\to D$ is a fibration, then $C$ has and $F$ preserves pullbacks if and only if each fiber of $F$ has pullbacks and the reindexing functors preserve them. Thus, the Grothendieck construction of any functor $D^{\mathrm{op}}\to \mathrm{Cat}$ which does not factor through $\mathrm{Cat}_{\mathrm{pullbacks}}$ will give a counterexample.

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  • $\begingroup$ Thanks Mike! Not only does this explain why I shouldn't find it in HTT, it also tells me the modified statement I should prove :). $\endgroup$ – David Carchedi Jun 27 '12 at 6:31

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