Daniel wrote:

> Another question I feel a bit ashamed to ask, but better late than never: Is it correct that


> could be drawn as


Yes! Nothing to be ashamed about there; that's a very important point and I was just too lazy to come out and say it. The bottom picture is perfectly correct. The top picture is a kind of abbreviation that turns out to be incredibly useful, for two main reasons:

1) We can omit drawing wires labelled by the unit object \\(I\\), roughly because \\(I \otimes x \cong x \cong x \otimes I\\) so putting \\(I\\) next to another object 'doesn't do anything'. (A more detailed explanation of why we can get away with this would take longer, and invoke Mac Lane's coherence theorem.)

2) Drawing the cap and cup the way we do makes the snake equations

seem 'intuitively obvious' and very easy to work with in calculations. This one of those examples of how a good notation does the thinking for you. Remember, the snake equations really say


are identity morphisms. But _that_ way of writing them makes them _not at all_ intuitively obvious!