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Can a category structure be defined on the collection $Adj(\mathbf C,\mathbf D)$ of all pairs of adjoint functors $$ (F\colon\mathbf C\to \mathbf D)\dashv (G\colon \mathbf D\to \mathbf C)$$ in such a way that the correspondence $\mathbf{Cat}\times\mathbf{Cat}\to \mathbf{Cat}\colon (\mathbf C,\mathbf D)\mapsto Adj(\mathbf C,\mathbf D)$ is functorial?

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Doubtful, since adjointness is not preserved by composition. However, you can make a category of adjoints – see [CWM, Ch. IV, §7]. – Zhen Lin Sep 8 '12 at 11:05
What if I take a different category structure on $\mathbf{Cat}$? Quite tautologically, if I take as morphisms $\mathbf C\to \mathbf D$ couples of adjoint functors between the two categories, I obtain a category $\mathbf{Cat}^A$; now suppose a morphism $\mathbf C\leftrightarrows \mathbf C'$ in $\mathbf{Cat}^A$ is given, then you can define at least a correspondence on objects $Adj(\mathbf C,\mathbf D)\to Adj(\mathbf C',\mathbf D)$. – Fosco Loregian Sep 8 '12 at 11:15
It is now simply a matter of deciding which is the "right" category structure on $Adj(\mathbf C,\mathbf D)$ in order to define a correspondence on arrows too. Which one of the two structures proposed in [CWM, IV, 7] is the most suitable? And these two are in some sense "compatible"? Are they completely different (I'm aware this is a different question: if you want I'll open a different topic)? – Fosco Loregian Sep 8 '12 at 11:17
The one in the main text (with "conjugate" pairs of natural transformations) is standard and can be made into part of a double category: see the examples here. It's not clear to me why you want to make $\textrm{Adj}(-, -)$ into a functor though. – Zhen Lin Sep 8 '12 at 11:45
No particular reason, just playing :) – Fosco Loregian Sep 8 '12 at 11:57

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