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When we are talking about a universal property, we say that there exists "a" unique morphism that satisfies blah blah blah. However, I, a non-native speaker of English, feel it should be "the" unique morphism since it is uniquely determined. I agree that it is natural to say "a" unique morphism up to isomorphisms*, but I am not sure why we say "a" unique morphism even when it is strictly unique. Is there any good mathematical explanation for this? (One attempt is that since the existence is not guaranteed, "the" sounds somehow awkward to native English speakers.) Any linguistic intuition will also be appreciated.

*Another question: I often see the notation "unique up to isomorphism", but why can we omit the article "a" in this case despite the fact that there can be multiple choices of isomorphisms?

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    $\begingroup$ "the" usually refers to something that has already been specified $\endgroup$ – J. W. Tanner Dec 22 '20 at 15:25
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    $\begingroup$ I suppose one says "a" unless the uniqueness has previously been established. That is to say, it is fine to speak of "the unique prime factorization of a natural number" as we have established that such a factorization is unique. But if we are in the process of demonstrating uniqueness then one says "a", as in "there is a unique polynomial with such and such properties." $\endgroup$ – lulu Dec 22 '20 at 15:25
  • $\begingroup$ Thank you very much for your clarifications. The comments helped me a lot. $\endgroup$ – P. Usada Dec 22 '20 at 15:30
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This is an interesting question as much about English language usage as mathematics.

First,

there exists the unique morphism

simply sounds wrong. That's reason enough to stick with the conventional "a". You don't want your readers or listeners to stumble over unfamiliar usage.

I can venture a reason. When you say "the" you are assuming existence, which is what your sentence is trying to assert. The proof usually runs along the lines "here is one, and in fact it's the only one". Using "a unique" somehow suggests the two separate steps.

As for

unique up to isomorphism

The same unfamiliarity holds. You could say

you could say

unique up to an isomorphism

but it wouldn't sound exactly right.

Grammatically, I read "up to isomorphism" as an adverb phrase modifying the adjective "unique" which in turn modifies the noun that names the essentially unique thing you are writing about.

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    $\begingroup$ I agree. "There exists the unique morphism" is incorrect in English. But explaining why is not about mathematics. $\endgroup$ – GEdgar Dec 22 '20 at 16:04
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In the context of universal properties a unique morphism may be different depending on the objects involved, so there can be many unique morphisms. For example in the category of rings $\mathbb{Z}$ is inital but the morphisms to $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ are each unique, but distinct. This makes it more reasonable to say that there exist a unique morphism since the unique morphism would imply there was only one morphism.

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