You may be interested in "[Functorial Approach to Graph and Hypergraph Theory][1]" by M. Schmidt. The paper looks at monoid actions as two object categories and then takes presheaves on these. It contains more than your average paper on categorical constructions on graphs. It is a generalization of the ideas contained in the paper "[Categorical Constructions in Graph Theory][2]" by Bumby and Latch. 


  [1]: https://arxiv.org/abs/1907.02574
  [2]: http://www.kurims.kyoto-u.ac.jp/EMIS/journals/IJMMS/Volume9_1/16.pdf