Normally you define a function to be a map on a set. But how about defining a function, in Category Theory, as a collection of arrows?
Take this cateogry
Objects: true, false.
true -> true false -> false true -> false false -> true
Here I had to specify FOUR arrows, but I could summarize this collection of arrows as TWO functions:
id(x) = x // replaces the first 2 not(x) = !x //replaces the second 2
If you want to maintain "arrowhood", which is more categoric, you could write a meta-arrow
idArrow(x) = x -> x notArrow(x) = x -> not(x)
Either way you compress information.
Is there a strandard way of expressing these meta-arrows? Is this even part of the theory?