I think you got your adjunctions backward John.

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}).\\]

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}).\\]