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Jencel Panic (Jul 18 2023 at 11:49):A terminal object in a category C is an object 1 of C satisfying the following universal property:
for every object x of C, there exists a unique morphism !:x→1
What is the reason for the word unique in that definition to be there?
Asking because it seems redundant at a first glance.
Morgan Rogers (he/him) (Jul 18 2023 at 11:52):In the category of sets, every non-empty set admits morphisms from every set. Only a -element set is the target of a unique morphism from every set.
Reid Barton (Jul 18 2023 at 12:02):("unique" means it's the only morphism from x to 1, in case that wasn't clear.)
Jencel Panic (Jul 18 2023 at 19:07):Thanks very much for the responses.