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This is a little surprising: it sounds like you're describing the Grothendieck construction, which is some kind of (infinity, 2)-categorical pullback, and not the (infinity, 1) categorical pullback. Are you sure your u, v are not required to be weak equivalences of some kind?
@Piotr, I don't think this is true: see my comment to Scott's answer. In fact, the way I want to think about this structure is as a log space underlying the real moduli space of smooth algebraic curves with marked points and real directions at all marked points. Whether or not this Kato-Nakayama realization happens to be locally constant over the base, it certainly does not admit a natural section.
