In "On numbers and games", Conway writes that the surreal Numbers form a universally embedding totally ordered Field. Later Jacob Lurie proved that (the equivalence classes of) the partizan games form a universally embedding partially ordered abelian group.
As far as I can tell from Lurie's paper, the latter means:
Let $\mathbb{U}$ denote the class of equivalence classes of partizan games. Furthermore let $S \subseteq S'$ be partially ordered abelian groups. Suppose that $\phi: S \rightarrow \mathbb{U}$ is an order-preserving homomorphism. Then there exists an order-preserving homomorphism $\phi': S' \rightarrow \mathbb{U}$ such that $\phi' \mid S = \phi$.
Now the thing is: I had before gotten to the (informal) conception that "universally embedding" in this context means that every (set-sized) partially ordered abelian group is isomorphic to some subgroup of $\mathbb{U}$, and vice versa that every totally ordered field is isomorphic to some subfield of $\mathbf{No}$. My question now is: is that even true? And what does this have to do with the statement of Lurie's paper?