On my favorite conventions for drawing the missing diagrams in Category Theory

Abstract

I used to believe that my conventions for drawing diagrams for categorical statements could be written down in one page or less, and that the only tricky part was the technique for reconstructing objects "from their names"... but then I found out that this is not so. This is an attempt to explain, with motivations and examples, all the conventions behind a certain diagram, called the "Basic Example" in the text. Once the conventions are understood that diagram becomes a "skeleton" for a certain lemma related to the Yoneda Lemma, in the sense that both the statement and the proof of that lemma can be reconstructed from the diagram. The last sections discuss some simple ways to extend the conventions; we see how to express in diagrams the ("real") Yoneda Lemma and a corollary of it, how to define comma categories, and how to formalize the diagram for "geometric morphism for children". People in CT usually only share their ways of visualizing things when their diagrams cross some threshold of of mathematical relevance - and this usually happens when they prove new theorems with their diagrams, or when they can show that their diagrams can translate calculations that used to be huge into things that are much easier to visualize. The diagrammatic language that I present here lies below that threshold - and so it is a "private" diagrammatic language, that I am making public as an attempt to establish a dialogue with other people who have also created their own private diagrammatic languages.