Universality of Barwick's unfurling construction

Given an $\mathrm{\infty }$-category $\mathcal{C}$ with pullbacks, its $(\mathrm{\infty },2)$-category $\mathbf{S}\mathbf{p}\mathbf{a}\mathbf{n}(\mathcal{C})$ of spans has the universal property of freely adding right adjoints to morphisms in $\mathcal{C}$ satisfying a Beck–Chevalley condition. We show that this universal property is implemented by an $(\mathrm{\infty },2)$-categorical refinement of Barwick’s \emph{unfurling construction}: For any right adjointable functor $\mathcal{C}\to {\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{\infty }}$, the unstraightening of its unique extension to $\mathbf{S}\mathbf{p}\mathbf{a}\mathbf{n}(\mathcal{C})$ can be explicitly written down as another span $(\mathrm{\infty },2)$-category, and on underlying $(\mathrm{\infty },1)$-categories this recovers Barwick’s construction. As an application, we show that the constructions of cartesian normed structures by Nardin–Shah and Cnossen–Haugseng–Lenz–Linskens coincide.