What is the meaning of collection in this context ? Is it here a synonym of set ?
Can someone please explain what the author means by "A moment's though shows that, as sets of functions, these two sets are the same."
Why are they the same ?
Here is the previous page for some more context:
The word 'collection', as I understand it, means something like 'things connected to each other in a way we intuitively understand but don't necessarily have a good mathematical definition for (and aren't going to worry about right now)'.
The comment about the two sets being the same is saying that, although we have two definitions of sets, the elements in them are exactly the same. If you take a map that is monotone by the presets definition, it is also monotone by the posets definition, and vice-versa.