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We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require locally (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category.arXiv:1909.07133v2 fatcat:6c6oqhtflrgwfovqpp3xiw3tae