  We generalize the First Reconstruction Theorem of Kontsevich and Manin in two
respects. First, we allow the target space to be a Deligne-Mumford stack.
Second, under some convergence assumptions, we show it suffices to check the
hypothesis of $H^2$-generation not on the cohomology ring, but on an any
quantum ring in the family given by small quantum cohomology.
