You wrote,

>This functor sends any pair \$$a, b\$$ consisting of an object \$$a\$$ in \$$\mathcal{A}\$$ and an object \$$b\$$ in \$$\mathcal{B}\$$ to the set

implying that,

\$b \text{ in } Obj(\mathcal{B}),\$

however next you wrote,

>$\mathcal{A}(F(a),b) .$

implying that,

\$b \text{ in } Obj(\mathcal{A}).\$