We prove an ambidexterity result for ∞-categories of ∞-categories admitting a collection of colimits. This unifies and extends two known phenomena: the identification of limits and colimits of presentable ∞-categories indexed by a space, and the ∞-semiadditivity of the ∞-category of ∞-categories with π-finite colimits proven by Harpaz. Our proof employs Stefanich's universal property for the higher category of iterated spans, which encodes ambidexterity phenomena in a coherent fashion.

I thank the anonymous referee for carefully reading the manuscript and for their helpful comments and suggestions. I am grateful to Lior Yanovski for numerous conversations about spans and ambidexterity over the years. I thank Bastiaan Cnossen and Thomas Blom for useful discussions, and Shai Keidar for his comments on the manuscript. I would like to thank the Max Planck--Weizmann joint postdoctoral program for its financial support, and the Max Planck Institute for Mathematics in Bonn for its hospitality during the workshop on "Unstable Homotopy Theory", during which a part of this manuscript was written.