General initiality for Hirschowitz–Maggesi signatures

Establish a general initiality theorem for the Hirschowitz–Maggesi notion of signatures, namely signatures defined using modules over monads in endofunctor categories, by proving that for every such signature there exists an initial object (initial model) in the corresponding category of models.

Background

The paper surveys and unifies three categorical approaches to initial semantics for programming languages with variable binding: the Fiore–Plotkin–Turi framework based on monoidal categories and functors with strength, the Hirschowitz–Maggesi framework using modules over monads (developed for endofunctor categories), and the Matthes–Uustalu formalism for constructing initial monads via Mendler-style recursion.

While the Hirschowitz–Maggesi approach provides a general and abstract definition of signatures and models, the abstract explicitly notes that there is currently no general initiality result for this notion of signature. Establishing such a theorem would complete the initial-semantics program for this framework, ensuring the existence of initial models and associated recursion principles for signatures defined via modules over monads.