What is Yoneda's Lemma a generalization of?
I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory.
Comments are welcome why the following candidates are good or bad ones.
Further examples are welcome!
Candidate #1: Axiom of extensionality (for sets)
A set is uniquely determined/can be recovered from its elements.
Candidate #2: Dedekind completions (for posets)
A completion of a poset S is the set of its downwardly closed subsets, ordered by inclusion. S is order-embedded in this lattice by sending each element x to the ideal it generates.
Candidate #3: Stone's representation theorem (for Boolean algebras)
Every Boolean algebra B is isomorphic to the algebra of clopen subsets of its Stone space S(B).
Candidate #4: Cayley's theorem (for groups)
Every group G is isomorphic to a subgroup of the symmetric group on G.