A Note on Finite-to-Infinite Extensions and Homotopy Invariance of Digraph Brown Functors

This paper develops extension theory for Brown functors in directed graph homotopy theory. We establish a systematic method for extending Brown functors from finite directed graphs to arbitrary directed graphs using inverse limits over finite subdigraphs. We prove that this extension is well-defined and preserves essential functorial properties. Additionally, we provide an alternative characterization of this extension through the Yoneda lemma, demonstrating how extended Brown functors can be naturally identified with sets of natural transformations from representable functors. This categorical perspective offers deeper theoretical insight into the structure of extended Brown functors and establishes important connections with classical representability theory, providing the technical foundation for Brown representability in directed graph theory.