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An adjunction is a relation between two functors. But I have come across something that feels like a “adjunction between natural transformations”. Does this concept exist?

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    $\begingroup$ You could identify a natural transformation with a functor, so... You can also generalize adjunctions to arbitrary 2-categories, so they don't need to be between functors. That said, I'm not aware of any concept called an "adjunction between natural transformations" and there no Google hits for that exact phrase, and since you haven't provided any description of what you mean by that, there's not much more that can be said. $\endgroup$ – Derek Elkins Mar 3 at 9:55
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    $\begingroup$ Perhaps you could give us an idea of what the concept you came across looks like $\endgroup$ – Max Mar 3 at 10:19
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    $\begingroup$ You may be interested in reading Higher transformations between natural transformations and so on and the comments there if you haven't already. $\endgroup$ – Ben Mar 3 at 10:33
  • $\begingroup$ In general, given two categories, $C$ and $D$, there is a category $Fun(C,D)$ whose objects are functors between $C$ and $D$, and morphisms are natural transformations between these functors. You can write down your favourite definition af adjunction in this category, and work out what this gives you for an adjunction bewteen two natural transformation $\endgroup$ – Thibaut Benjamin Mar 3 at 10:52

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