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

Given an $\mathrm{\infty }$-category $C$ equipped with suitable wide subcategories $I,P\subset E\subset C$, we show that the $(\mathrm{\infty },2)$-category $\text{S}{\text{PAN}}_2(C,E{)}_{P,I}$ of higher (or iterated) spans defined by Haugseng has the universal property that 2-functors $\text{S}{\text{PAN}}_2(C,E{)}_{P,I}\to \mathbb{D}$ correspond precisely to $(I,P)$-biadjointable functors $C^{\text{op}}\to \mathbb{D}$, i.e. functors $F$ where $F(i)$ for $i\in I$ admits a left adjoint and $F(p)$ for $p\in P$ 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 $C^{\text{op}}\to \text{CAlg}(\text{Cat})$.