Cantor–Bernstein type theorem for locally constrained graph homomorphisms

We show that the simultaneous existence of a single locally surjective graph homomorphism between a graph GG and a connected and finite graph HH together with some locally injective homomorphism between the same pair of graphs assures that both homomorphisms are locally bijective.We give a short proof of this assertion which unifies previously known partial results of this form. We utilize the notion of universal cover, and relate its properties to the notion of degree refinement, which was used as a principal tool in other works.