@Peter – it might be easier to first consider just one category \\(\mathcal{C}\\) and try to build a functor that sends a pair of objects \\((X, Y)\\) to the set of morphisms from \\(X\\) to \\(Y\\), namely \\(\mathcal{C}(X, Y)\\).

If you think about how this might works on pairs of morphisms \\((f, g)\\) you'll notice it has to be contravariant in the first variable and covariant in the second one. Hence it's a functor from \\(\mathcal{C}^{op} \times \mathcal{C}\\) to \\(\textbf{Set}\\).

The functors \\(\mathcal{B}(F(-), -)\\) and \\(\mathcal{A}(-, G(-))\\) are variants of this construction.