This is an embarrassing question, because I learned about this theorem in basic analysis, but haven't realized that I don't really understand its statement until now.
Anyway, it's a famous result that "up to isomorphism, $\mathbb{R}$ is the only Dedekind-complete ordered field". Sources, e.g. (1)(2)(3)(4)
Question: isomorphism in which category?
Isomorphic as fields? Isomorphic as ordered fields? Isomorphic as rings? Isomorphic as sets? (clearly not) Isomorphic as topological spaces? (also clearly not, but you get my point)