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Why not just say that the cone $U \leftarrow U \rightarrow U$ given by identities is a product?
I was just making an (IMO) minimal adjustment to the existing language. I don’t really know what precisely the original wording wanted to emphasize. I just wanted to reword it to forestall any reactions wondering about $U \times U$ not existing that might slow a reader down.
umm
An object $U$ in a category $C$ is subterminal or preterminal if any two morphisms with target $U$ and the same source are equal. In other words, $U$ is subterminal if for any object $X$, there is at most one morphism $X\to U$.
If C has a terminal object 1, then U is subterminal precisely if the unique morphism U→1 is monic, so that U represents a subobject of 1; hence the name “sub-terminal.”
like anything associated with limits while the objects are unique the morphisms from or to them are not equal or unique but only unique up to isomorphism,
Re #6: The quoted text looks correct. The sentence after the quote seems to have the roles of objects and morphisms mixed up.
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