@PeterJones, I was confused at first by that too, but I think it's making sense to me now.

It seems that part of the idea is that you're supposed to take this very zoomed out abstract view, where you don't care so much about the individual objects in a category (in this case, sets), but you care more about the morphisms, or the relationships between them.

With that (admittedly vague) idea in mind, consider a set with only one element in it, and all the morphisms (functions) it has to a set with n elements in it. How many possible different functions are there from a set with one element to a set with n elements?

The answer is that there are n such functions -- one for each element in the target set. So you could say that all the morphisms (functions) from the 1 element set to the n element set "range over" the elements in that set.

Or, looking at it a different way, if you want to represent the idea of "picking out" a single element from a set S, you could think of that as the same thing as choosing one function from among all the possible functions from a one element set to S.

Similarly, as I think Brendan mentioned more briefly, if you wanted to choose a pair of elements in a set S, you could think of that as the same thing as choosing one function from a two element source set to S as the target set (because each possible pair of elements in S -- including pairs that have the same element twice -- is a possible mapping/function from the two source elements in the two-element set).

As for why it's natural to think of things this way. I'm not really sure yet, as I'm just learning this stuff, but I think it comes from having a very morphism-centric point of view, or something like that.

It seems that part of the idea is that you're supposed to take this very zoomed out abstract view, where you don't care so much about the individual objects in a category (in this case, sets), but you care more about the morphisms, or the relationships between them.

With that (admittedly vague) idea in mind, consider a set with only one element in it, and all the morphisms (functions) it has to a set with n elements in it. How many possible different functions are there from a set with one element to a set with n elements?

The answer is that there are n such functions -- one for each element in the target set. So you could say that all the morphisms (functions) from the 1 element set to the n element set "range over" the elements in that set.

Or, looking at it a different way, if you want to represent the idea of "picking out" a single element from a set S, you could think of that as the same thing as choosing one function from among all the possible functions from a one element set to S.

Similarly, as I think Brendan mentioned more briefly, if you wanted to choose a pair of elements in a set S, you could think of that as the same thing as choosing one function from a two element source set to S as the target set (because each possible pair of elements in S -- including pairs that have the same element twice -- is a possible mapping/function from the two source elements in the two-element set).

As for why it's natural to think of things this way. I'm not really sure yet, as I'm just learning this stuff, but I think it comes from having a very morphism-centric point of view, or something like that.