Sound interpretation of N in the proposed comonadic modality semantics

Ascertain whether, in the proposed comonadic modality semantics based on the category S⋉R (with adjunctions inducing the dependent right adjoint comonadic modality +* ∘ (-,0)^* on presheaf categories), the natural numbers object can be soundly interpreted as the representable presheaf y_{(0,1)}. If such an interpretation fails, identify the necessary modifications, such as changes to the base categories, the sheafification step, or transitioning to MATT, to obtain a sound interpretation.

Background

To extend the system with a comonadic modality enabling more expressive definitions while preserving conservativity over primitive recursion, the authors sketch a semantics using the category S of arities and functions and the semidirect product S⋉R. This yields adjunctions that induce a dependent right adjoint comonadic modality in presheaf categories.

A key unresolved step in validating this semantics is interpreting the natural numbers object. The authors note that it remains to be checked whether N can be modeled as the representable presheaf at (0,1) under the Yoneda embedding in this setting; otherwise, structural adjustments (e.g., to the base category, sheafification, or moving to MATT) may be required.