What is the right category in which to think of adjoints? I still find adjoints tricky to wrap my head around. I think this is possibly because my intuition for most constructions I've seen in category theory (functors, natural transformations, limits / colimits) has fundamentally to do with my geometric intuition for directed graphs and diagrams.
I think that I may be finding adjoints tricky because their data seems to twist several different categories around. To be more concrete with my question:
C and D are categories, and $F : C \to D$ and $G: D \to C$ are left and right adjoint functors. Is there a category in which the natural bijection between $Mor(F(A),B)$ and $Mor(A,G(B))$ naturally lives?

Yes. It lives in the category of functors $C^{op} \times D \to \text{Set}$; this is the category of profunctors from $C$ to $D$. 
