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?
Actually the isomorphisms are a little bit more than natural transformations. Suppose the isomorphisms are denoted by $\eta_{A,B}:\mathscr{B}(F(A),B)\to\mathscr{A}(A,G(B))$, then for fixed $A\in\mathscr{A}$, $\eta_{A,-}$ is a transformation $\mathscr{B}(F(A),-)\Rightarrow\mathscr{A}(A,G(-))$, and similarly for fixed $B\in\mathscr{B}$, $\eta_{-,B}$ is a transformation $\mathscr{B}(F(-),B)\Rightarrow\mathscr{A}(-,G(B))$. The "wired" statements are just saying these.
These conditions make $\eta_{-,-}$ a natural transformation between bifunctors.