Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually interesting. To prove a property $P$ (largeness, linear over $\mathbb{Z}$, virtual fibering, LERF, virtually biorderable etc.) for a class of objects $X$ (*e.g.* fundamental groups of finite volume hyperbolic $3$-manifolds), embed each object of $X$ nicely in an object of a `universal receiver' class of objects $R$ (Right-Angled Artin Groups, or RAAGs), each of which are simple and has good properties. The existence of such an embedding **in itself** implies property $P$ which you are interested in, maybe with some additional effort (Agol's fibering theorem, tameness, etc.).

Proving a mathematical statement in this way makes a lot of sense, but I don't recall having seen this proof pattern before anywhere else in mathematics. Well, that's not entirely true- Cayley's Theorem that a group embeds in a permutation group has some corollaries (*e.g.* Given a group $G$ and subgroup $H$ with $[G:H]=n$, there exists a exists a normal subgroup $N$ of $G$, with $N\subseteq H$ such that $[G:N]|n!$).

Question: Which other conjectures, that objects in a class $X$ have a property $P$, have been proven by embedding objects in $X$ nicely as subobjects of objects in a universal receiver $R$ whose good properties imply $P$ for objects in $X$?

For a compelling example, it would have to be difficult to prove $P$ for objects in $X$ in any other way. For an even more compelling example, the universal receiver $R$ would be surprising (RAAGs are a surprising universal receiver, I think).