What are the legitimate morphisms between algebraic graph rewrite rules? The question is complicated by the diversity of approaches. From the familiar Double-Pushout (DPO) to the more recent PBPO and many others, the rules have different shapes, semantics (defined by direct transformation diagrams) and even matchings. We propose to represent these approaches by categories of rules, direct transformations and matchings related by functors in Rewriting Environments with Matchings. From these we extract a so-called X-functor whose properties are key to make rule morphisms meaningful. We show that these properties are preserved by combining approaches and by restricting them to strict matchings.