Last week I attended a Category Theory meetup, and as usual, it looks like "standard" notations were chosen to be maximally confusing. And then there's implicit lifting everywhere. 1/n
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Some day, I'll understand enough Category Theory to do my own variant with blackjack and hookers—and some coherent notation. And either all the arrows will go from right to left, or function application will be postfix by default.
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The most elementary result of Category Theory, Yoneda's Lemma, requires three levels of arrows, relating internal arrows to external ones that preserve two levels of arrows. Nothing is interesting below three levels: 1. Morphisms, 2. Functors, 3. Natural Transformations.
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There seems to be no convention for diagrams over multiple levels of arrows. Wikipedia's page for the Yoneda Lemma has a semi-nice one with one square inside another, but only position relates two levels, and for more complex diagrams, you'll need at least a third dimension.
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