example, ${\operatorname}{Span}_k({\mathcal}{X},X_{\bullet,\dots,\bullet})$ itself forms a stack $${\mathcal}{X}^{op}\xrightarrow{U\mapsto {\operatorname}{Span}_k({\mathcal}{X}_{/U},U\times X_{\bullet,\dots,\bullet})}{CSS}_k(\hat{\mathscr{S}})$$

over ${\mathcal}{X}$. As a second example, taking ${\mathcal}{X}={\operatorname}{dSt}_{{\mathbb}{K}}$ to be derived stacks over a field ${\mathbb}{K}$ of characteristic zero, and $X_{\bullet,\dots,\bullet}$ to be trivial, the fact that forms a stack implies that the *derived* composition of spans depends continuously (algebraically, in fact) on the spans involved.

Our motivation for these results comes from mathematical physics: the success of the *Lagrangian Creed*:

*“everything is a Lagrangian correspondence”[^2],*

places Lagrangian correspondences between symplectic manifolds at the centre of classical mechanics. Lagrangian correspondences have two major flaws however: **firstly** they fail to compose in general, i.e. given two Lagrangian correspondences $$U\overset{L}{\nleftarrow} V\text{ and }V\overset{L'}{\nleftarrow} W,$$ their set theoretic composite $$\label{eq:LagComp}U\overset{L\circ L'}{\nleftarrow} W,$$ often fails to be smooth, and - **secondly** - when the composite exists as a Lagrangian correspondence, it may not depend continuously on $L$ and $L'$.

The first of these issues was essentially resolved by Pantev, Toën, Vaquié, and Vessozi [@Pantev:2011uz], and Calaque [@Calaque:2013un] using derived geometry. Building upon this, Haugseng [@Haugseng:2014vw] then gave an embedding of Weinstein’s symplectic ‘category’ [@Weinstein82] whose morphisms are the Lagrangian correspondences, as a subcategory of ${\operatorname}{Span}_1({\operatorname}{dSt}_{{\mathbb}{K}}, {\mathcal}{A}^2_{{\operatorname}{cl}})$, spans of derived stacks with local systems valued in 