To put it as a single question: why is it that composition of arrows in a category should be associative? Where do these sorts of axioms come from, really?
I think the answer is basically this diagram:
Even if composition of arrows itself is assumed weakly enough that all you get is a 2-cell mediating the connection between the path that you are composing and the direct path between source and target, you still are able to compose these 2-cells to get from one composition order to another, and this feels like an essentially geometric fact somehow. Of course you can get back, too because you know that that 2-cell is invertible. (indeed I am already using both directions of it in the above diagram at different points) And the identity laws are essentially just nullary versions of associativity.
The annoying thing is how difficult it is to formalize n-dimensional cell complexes in the appropriate way. This is also the part I'm most certain has been done a million times already, but it seems like good practice to bash my brain against it awhile.