Full faithfulness of the nerve functor within the double category C

Establish, entirely within the formalism of the double category C of categories, cofunctors, familial functors, and transformations, a proof that the nerve functor N_m from the category of algebras for a familial monad m on a copresheaf category c to presheaves on the associated theory category Θ_m (defined by N_m(A)(M) = Hom_c(m[M], A)) is fully faithful, thereby reproducing Weber’s Nerve Theorem within C.

Background

The paper develops a categorical description of theory categories and nerve functors using the double category C constructed from polynomial functors, comonoids, and bicomodules. Within this framework, theory categories Θ_m are obtained by a right coclosure (left Kan extension) construction, and the nerve of an m-algebra is modeled via a coalgebra structure in C.

Weber’s Nerve Theorem classically proves that the nerve functor N_m: Alg(m) → PSh(Θ_m) is fully faithful. While the authors reconstruct Θ_m and the nerve functor in C, they explicitly note that they have not yet expressed the proof of full faithfulness in this formalism. Providing a C-internal proof would complete the program of rederiving the nerve theorem entirely within the polynomial/double-categorical setting, clarifying the canonicity and universality of the construction in C.