Intuitive introduction to Abstract Nonsense

TL;DR John Wentworth has written a great introduction to category theory for someone who wants to do something useful with categories. This post has a different goal: Assuming one is curious about the useless manifestations of category theory, what is a quick intuition to get started? This post's proposal: the game Recover the Labels.The arcade machine The "game" is played on an arcade machine with three screens.Left screen Through the left screen we can see sets, each of which takes one specific location over a 2D[1] background. The game lets us scroll and see other locations with other sets. Sets are unordered lists of things, each and every set is out there somewhere to be found if we scroll enough. We have e.g. the set "Countries in North America", which we can also write as {Canada, USA, Mexico}, and we have a set of two websites {LessWrong, AstrologyToday}, and we also have more typical sets, like the set of all natural numbers, the set of all real numbers, etc. There are also peculiar sets like the set of all natural numbers except the number 12, or the set which has as members "Canada", "AstrologyToday" and all real numbers greater than 13.The left screen shows not only sets, but also functions between sets. A function between two sets A and B is a rule that assigns to each element of A exactly one element of B. We can e.g. imagine that on different days of this year, LessWrong or AstrologyToday was the most popular website in Canada, USA and Mexico. If we make a list which shows the most popular website for each of those three countries in one specific day, we've defined a function.We can also define functions in the other direction: We imagine days in which both LessWrong and AstrologyToday had on their main page a story about Canada or USA or Mexico (but only one of those countries!). If we make a list which shows the featured country for each of those two websites in one specific day, we have defined one function.Exercise for the reader: Show that there ar