The mnemonic value of "operators" as reminders of properties such as associativity or commutativity interestingly extends to diagrammatic reasoning. A diagram

The mnemonic value of "operators" as reminders of properties such as associativity or commutativity interestingly extends to diagrammatic reasoning. A diagram is really just a generalized expression, and this becomes quite useful when one has to deal with more than one "type" or "domain" of operation, but in a consistent way that preserves the compositionality-like properties OP talks about. A recent book exploring this topic is "Seven Sketches in Compositionality; An Invitation to Applied Category Theory" https://arxiv.org/abs/1803.05316 https://arxiv.org/abs/1803.05316 . (Despite the obvious reference to CT in the title, the work is quite accessible and the math involved is not much more complicated than that found in the linked blogpost. Importantly, and perhaps unlike some people in other programming-language communities, the author does not assume any pre-existing knowledge; the work is rather about using concrete, real-world examples to gently guide the reader's intuition.)