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A cospan is two morphisms having the same codomain. A pullback is a limit of a cospan. If instead of having two morphisms, we have a set of morphisms having the same codomain, is there a name corresponding to "cospan" and a name corresponding to "pullback" (as we have with "discrete diagram" and "Cartesian product")?

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These are commonly called wide pullbacks. As far as I can tell, this terminology is due to Definition 1.2.3 of Taylor's Quantitative domains, groupoids and linear logic. The term wide cospan for the analogous diagram is in use, but appears to be slightly less common.

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