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?