Equivalence between loose right modules and double operad algebras

Establish an equivalence between the 2-category of symmetric monoidal loose right modules over isofibrant double categories and pseudo symmetric monoidal pseudo double functors whose unitors and laxators are companion commuter transformations, and the 2-category of algebras for isofibrant double operads with all tensors and pseudo-morphisms whose universal comparison map between tensors is companion to an isomorphism, thereby clarifying the precise relationship between the double operadic and module-based formulations of categorical systems theory.

Background

The paper adopts symmetric monoidal loose right modules over symmetric monoidal double categories as the primary framework for categorical systems theory, rather than algebras for double operads. In 1-categorical settings, such module-like structures correspond closely to operad algebras via lax monoidal functors, but in the double categorical setting this correspondence is more subtle.

To reconcile these two perspectives, the authors propose a specific conjectural equivalence between a 2-category built from loose right modules (with constraints on unitors and laxators) and a 2-category of algebras for isofibrant double operads (with tensor conditions and a companion property on universal comparison maps). Proving this equivalence would unify operadic and module-based approaches to systems composition in the double categorical landscape.