Leinster defines the notion of adjoint functor in the following way: click (or see pp. 41-42). He then remarks that in the future, the naturality conditions will be interpreted as the commutativity of some square (Remarks 2.1.2). What he means is Remark 4.1.24: click (or see pp. 91-92).
My understanding is that the naturality conditions can also be interpreted as the commutativity two squares (or rather infinite number of squares of two types), one corresponding to the fact that the functors $$\mathscr B(F(A),-):\mathscr B\to\mathbf{Set}\text{ and } \mathscr A(A,G(-)):\mathscr B\to\mathbf{Set}$$ are naturally isomorphic (for every fixed $A$), and the other corresponding to the fact that the functors $$\mathscr B(F(-),B):\mathscr A\to\mathbf {Set} \text{ and } \mathscr A (-,G(B)):\mathscr A\to\mathbf {Set}$$ are naturally isomorphic (for every fixed $B$).
Is this statement right? And so is the commutativity of these two series of squares equivalent to the commutativity of the square from Remark 4.1.24? If so, is there an easy way to see that the isomorphism between the above two functors is equivalent to the isomorphism between the functor from Remark 4.1.24, or does that require a careful bothersome proof in any case?
The category of categories is cartesian closed; it means that $$ {\bf Cat}(\mathscr{A}\times\mathscr{B},\mathscr{C}) \cong {\bf Cat}(\mathscr{B},\mathscr{C}^{\mathscr{A}}) \cong {\bf Cat}(\mathscr{A},\mathscr{C}^{\mathscr{B}}) $$ where at the codomain there is the category of functors $\mathscr{C}^{\mathscr{A}} = {\bf Cat}(\mathscr{A},\mathscr{C})$ and similarly for $\mathscr{C}^{\mathscr{B}}$; this not only means that there is a bijection between the two sets/classes, but also that these three categories of functors are isomorphic; so, there are fully faithful and bijective on objects functors linking the three categories of functors.
The presence of such a fully faithful morphism entails that a natural transformation $\eta : F \Rightarrow G : \mathscr{A}\times\mathscr{B} \to\mathscr{C}$ is indeed natural if and only if the corresponding natural transformations $\dot\eta : F(A,-)\Rightarrow G(A,-) : \mathscr{B} \to \mathscr{C}^{\mathscr{A}}$ and $\ddot \eta : F(-,B)\Rightarrow G(-,B) : \mathscr{A} \to \mathscr{C}^{\mathscr{B}}$ are natural.