Categorical Ambidexterity

We prove an ambidexterity result for $\mathrm{\infty }$-categories of $\mathrm{\infty }$-categories admitting a collection of colimits. This unifies and extends two known phenomena: the identification of limits and colimits of presentable $\mathrm{\infty }$-categories indexed by a space, and the $\mathrm{\infty }$-semiadditivity of the $\mathrm{\infty }$-category of $\mathrm{\infty }$-categories with $\pi$-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.