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We study four types of (co)cartesian fibrations of oo-bicategories over a given base B, and prove that they encode the four variance flavors of B-indexed diagrams of oo-categories. We then use this machinery to set up a general theory of 2-(co)limits for diagrams valued in an oo-bicategory, capable of expressing lax, weighted and pseudo limits. When the oo-bicategory at hand arises from a model category tensored over marked simplicial sets, we show that this notion of 2-(co)limit can be calculated as a suitable form of a weighted homotopy limit on the model categorical level, thus showing in particular the existence of these 2-(co)limits in a wide range of examples.