Universality of span 2-categories and the construction of 6-functor formalisms

Given an ∞-category C equipped with suitable wide subcategories I, P ⊂ E⊂ C, we show that the (∞,2)-category S₂(C,E)P,I of higher (or iterated) spans defined by Haugseng has the universal property that 2-functors S₂(C,E)P,I → 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).