Boolean networks (BNs) have become quite popular since their proposal as models of network regulation. Every BN can be decomposed into coordinate functions and thus defines a signed, directed graph, called a wiring diagram, that describes the variable dependencies. We will explore what it means for two BNs to be equivalent and how to define a structure-preserving map between them. In particular, maps that are topologically conjugate or semi-conjugate need not preserve locality of the functions or the wiring diagram. We will illustrate this with examples and non-examples, involving commutative diagrams with extra structure. Finally, we will discuss how to define a category of Boolean networks and explore some of its basic properties.