This post is a basic introduction to the idea of Categorical Graphs, or just the standard theory of Graphs, developed through the lens of Category Theory, and generalized in the obvious ways that follow through that lens. This is just an introduction mainly because this theory doesn’t seem to have been developed very far yet, and I haven’t been able to develop it that much independently so far. I think I have a good grasp on some very basic ideas here, and wanted to present them while they’re still fresh in my head.