Are Free Groups the "Smallest Group" Containing their Generators

Question

I apologize if this is a duplicate; I was not sure how to search for this.

When I say "the smallest group" I mean unique up to isomorphism of course.

Specifically, is "the smallest group containing its generators" an intuitive way of interpreting the universal property given on Wikipedia? The homomorphism $\varphi$ being an "embedding" of sorts?

With set systems, for example $\sigma-$algebras or topologies, whenever we say "the X generated by S" we usually mean equivalently "the smallest X containing S" (in some sense).

So essentially my question is this: is the intuition which is valid for set systems extensible to the thinking behind free algebraic objects?

Or at least to free groups, since the intersection of two groups is again a group, the same way the intersection of two topologies is a topology or the intersection of two $\sigma-$algebras is a $\sigma-$algebra.

Answer 1

I believe you have things backwards: as the comments indicate, $F_S$ is in a sense the $\textit{largest}$ group for a given set of generators. If $S$ the set of generators for a group $G$, the map from $F_S \to G$ is surjective and hence any group $G$ may be identified with a quotient of $F_S$.

If by "smallest" you mean least structure, then this is correct as we have no relations.