In the recent months I've come across a phenomenon which seems to come up in several areas of algebra making me wonder if there's a larger concept behind it, which I just fail to grasp. Namely, embedding large classes of algebraic structures in symmetric ones. For example, in the case of finite groups, a theorem by Cayley tells us that we can always embed them in symmetric groups, for Lie algebras we know that they are always contained in their universal enveloping algebra (the associated graded of which is the symmetric algebra) and in the case of representation theory of the general linear group we have Schur-Weyl-duality.
So my question is: Are these instances of a more general concept? Could these results be categorified for, say, fusion categories and symmetric ones? I realize this is a very vague question but I thought I'd give it a shot.