2-functoriality of the state category construction

Determine whether the state category construction that maps each copy-discard category C to its state category C^S, and each functor F: C → D to a functor F^S: C^S → D^S, extends to a 2-functor by assigning to every natural transformation η: F ⇒ G a corresponding natural transformation η^S: F^S ⇒ G^S compatible with the state-category state transitions and output functions; if so, construct η^S or characterize necessary and sufficient conditions for its existence.

Background

The paper defines the state category C^S for a copy-discard category C and shows how a functor F: C → D induces a functor F^S: C^S → D^S. This establishes a functorial action on objects and 1-morphisms (functors).

The authors then attempt to lift this to 2-morphisms by defining, for a natural transformation η: F ⇒ G, a candidate η^S: F^S ⇒ G^S. They identify a technical obstacle in constructing the output functions for η^S, specifically the lack of natural choices for morphisms of type G(X_i) → G(Y_i) needed to complete the construction.

Resolving whether such an assignment exists would establish whether the state category construction is genuinely 2-functorial and clarify how natural transformations interact with state transitions and generated output morphisms.