Representing a logic signature with a digraph

Question

I'm researching Monoidal Categories and became interested in the concept of signatures. In Peter Selinger's survey we can find the definition:

A simple (categorical) signature Σ consists of a set Σ0 of object variables, a set Σ1 of morphism variables, and a pair of functions dom, cod:Σ1→Σ0. Object variables are usually written A, B, C, . . ., morphism variables are usually written f, g, h, . . ., and we write f : A → B if dom(f) = A and cod(f) = B.

To program with Monoidal Categories I want a plain-text syntax, such as most programming languages. I came across YAML which has a graph-based spec. I had a pretty simple but powerful idea. I couldn't find any treatment of this topic so I reach out to you for guidance.

Does the following definition support arbitrary multi-sorted signatures based on a digraph, and YAML by extension?

Given a graph G, a signature Σ consists of Σ0=Σ1=graph nodes, dom=predecessors, cod=successors.

Thank you.

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See also: The definition of a free monoidal category

Answer 1

This isn't sufficiently arbitrary. For example non-symmetric signatures.

For a complete treatment see Section 1.6 Fibrations of signatures in Bart Jacob's Categorical Logic and Type Theory.