  We now have a wide range of proof assistants available for compositional
reasoning in monoidal or higher categories which are free on some generating
signature. However, none of these allow us to represent categorical operations
such as products, equalizers, and similar logical techniques. Here we show how
the foundational mathematical formalism of one such proof assistant can be
generalized, replacing the conventional notion of string diagram as a
geometrical entity living inside an n-cube with a posetal variant that allows
exotic branching structure. We show that these generalized diagrams have richer
behaviour with respect to categorical limits, and give an algorithm for
computing limits in this setting, with a view towards future application in
proof assistants.
