Left adjoint for the diagram contexts functor

Determine whether the functor that forms diagram contexts in strict monoidal categories, Ctx: MonCat → MultiCat (as defined via the formation of diagram contexts over a monoidal category), admits a left adjoint; construct such a left adjoint if it exists, or prove that no such left adjoint exists, thereby clarifying whether a direct “contour” for diagram contexts can be defined analogously to the contour adjoint for spliced arrows.

Background

Melliès and Zeilberger showed that the splicing construction on categories has a left adjoint (the contour), which is central to a generalized Chomsky–Schützenberger theorem in categorical settings. In extending this framework to monoidal categories, the paper defines diagram contexts and proves that their formation extends to a functor Ctx. However, the authors note that it is not clear whether Ctx itself has a left adjoint.

To proceed with their main results, the authors introduce the multicategory of raw optics and prove an adjunction between the optical contour and raw optics, thereby sidestepping the uncertainty about a direct adjoint for diagram contexts. Whether a left adjoint exists directly for Ctx remains unresolved and is explicitly marked as unclear in the text.