Cartesian multicategories as retromodels

Let $\ifstrequal{S}{1}{\mathds{1}}{\mathbb{S}}$ be the double theory of symmetric multicategories, let $\ifstrequal{S}{1}{\mathds{1}_{pan}}{\mathbb{S}\mathsf{pan}}$ be the double category of spans, and suppose that a double category $\ifstrequal{D}{1}{\mathds{1}}{\mathbb{D}}$ admits companions. Write $\ifstrequal{D}{1}{\mathds{1}}{\mathbb{D}}^\mathrm{rev}$ for the double category with the same objects, arrows, and proarrows as $\ifstrequal{D}{1}{\mathds{1}}{\mathbb{D}}$, together with its retrocells.

Cartesian multicategories as retromodels. The category of cartesian models is the category of retromodels of $\ifstrequal{S}{1}{\mathds{1}}{\mathbb{S}}^\mathrm{op}$ in $\ifstrequal{S}{1}{\mathds{1}_{pan}}{\mathbb{S}\mathsf{pan}}$, equivalently the category of cartesian lax double functors and strict tight transformations

$$\ifstrequal{S}{1}{\mathds{1}}{\mathbb{S}}^\mathrm{op}\to \ifstrequal{S}{1}{\mathds{1}_{pan}}{\mathbb{S}\mathsf{pan}}^\mathrm{rev}.$$

This identifies the contravariant action associated to cartesian multicategories with retrocells in the span double category. The source notes that details remain to be checked, so the conjecture is presented as a proposed axiomatization rather than an established equivalence.