The technique of equipping graphs with an equivalence relation, called equality saturation, has recently proved both powerful and practical in program optimisation. However, such structures, which are called e-graphs, apply only to certain kinds of theories, namely to algebraic theories where the generators have a single output which is inherent to what an abstract syntax tree is. The theories where the generators can have many inputs, as well as many outputs, are called monoidal theories. These theories have a wide area of applications, including quantum computing and digital circuits, which would benefit from having a corresponding e-graph machinery for them. We provide a generalisation of e-graphs for arbitrary monoidal theories through building a categorical construction. We argue that e-graphs themselves can be seen as a specific instance of this construction when applied to Cartesian monoidal theories. This not only paves the way for new applications but also lays a solid foundation for reasoning about e-graphs.