• Isomorphism (≅): twins — perfectly matching via some correspondence
• Natural isomorphism: indistinguishable twins — the correspondence is essentially forced
• Equivalence (≃): different people playing the same role

(Just an analogy, of course.)

So the term free functor comes from those classical Set-based examples, but in general a left adjoint need not look “free” at all.

For example:

• Set → Top gives the discrete topology
• Grp → Ab gives the abelianization \(G/[G,G]\)

These constructions satisfy the universal property of a left adjoint, but they don’t feel “free” in the same way as free groups or free vector spaces.