@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.