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Eric: How is this (note: terminal object) the universal cone over the empty diagram?
Toby: It seems to me that this is really a question about terminal objects in general than about terminal objects in . A cone over the empty diagram is simply an object, and a morphism of cones over the empty diagram is simply a morphism. A universal cone over a diagram is a cone over such that, given any cone , there is a unique cone morphism from to . So a univeral cone over the empty diagram is an object such that, given any object , there is a unique morphism from to . In other words, a universal cone over the empty diagram is a terminal object.
I don't see the point of the last paragraph before this query box. Already at the end of the previous paragraph, we've proved that is a terminal object, since there is a unique function (morphism) to from any set (object) . It almost looks like you wrote that paragraph by modifying the paragraph that I had written in that place, but that paragraph did something different: it proved that was unique. Apparently, you thought that this was obvious, since you simply added the word ‘unique’ to the previous paragraph.
Alternatively, if you want to keep a paragraph that proves unicity, then you can remove ‘unique’ and rewrite my original unicity proof in terminology more like yours, as follows:
Now let be any function. Then
for any element of , so .
Eric: Thanks Toby. I think what I’m looking for is a way to understand that a singleton set is the universal cone over the empty diagram. All these items should be seen as special cases of limit. Unfortunately, I don’t understand limit well enough to explain it. In fact, that is one of the reasons to create this page, i.e. so that I can understand limits :)
The preceding paragraph was my attempt to make it look like a limit, but I obviously failed :)
Ideally, this section would show how terminal object is a special case of limit somehow.