Given an -category C equipped with suitable wide subcategories I, P E C, we show that the (, 2) -category S₂ (C, E) , ₈ of higher (or iterated) spans defined by Haugseng has the universal property that 2-functors S₂ (C, E) , ₈ D correspond precisely to (I, P) -biadjointable functors Cᵒp D, i. e. functors F where F (i) for i I admits a left adjoint and F (p) for p P admits a right adjoint satisfying various Beck-Chevalley conditions. We also extend this universality to the symmetric monoidal and lax symmetric monoidal settings. This provides a conceptual explanation for - and an independent proof of - the Mann-Liu-Zheng construction of 6-functor formalisms from suitable functors Cᵒp (Cat).
Cnossen et al. (Sun,) studied this question.
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