In what sense is the uniqueness of left adjoint isomorphism 'canonical'

Question

In my category theory course, Peter Johnstone has written that for any two left adjoints $F$, $F'$ "there is a canonical natural isomorphism $F \to F'$"
Explicitly, this isomorphism is that for any $A$, $(FA, \eta_A)$ and $(F'A, \eta'_A)$ are both initial objects of $A\downarrow G$ so are uniquely isomorphic, and this gives the $A$ component of a natural isomorphism.
That this isomorphism of initial objects surely doesn't imply that there is a unique natural isomorphism $F \to F'$ though, so I was wondering in what sense it would be unique (if it is) or canonical. Is there some sort of higher naturality going on?
Intuitively I would think it is unique, since being a left adjoint is sort of a property rather than a structure, so isomorphisms shouldn't carry any data.