Here is how the definition of adjoint functor reads on page 41 of Leinster's "Basic Category Theory":
I understood this part thanks to Definition of Adjunction in Category Theory. It clarifies that the adjunction is a natural isomorphism between certain functors. However, as Remark 2.1.2 on the book seem to suggest, Leinster is postponing this interpretation to Chapter 4 and transcribing the definition of natural isomorphism for these functors:
Why is he doing so?