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I have the following question.

Suppose I have a functor $F\colon C\to D$ between two categories. I would like it to have an adjoint (say, right), but it doesn't. Is there a way to define a "best approximation" to its adjoint?

An obvious way is to define $G\colon D\to C$ be a functor with a natural transformation, say, $\alpha_G\colon Hom(F(-),-)\to Hom(-,G(-))$ such that $G$ is universal. I mean, such that for any other $H\colon D\to C$ and $\alpha_H\colon Hom(F(-),-)\to Hom(-,G(-))$, $\alpha_H$ will factor through $\alpha_G$.

I guess we can give similar definition with arrows $\alpha_H$ going in the opposit direction. The problem is I am not sure if such definition will be at all useful. It feels like the problem of finding "approximations to an adjoint functor" is similar to finding factorization of one functor through another (where the best solution is given by Kan extension). But I am not sure if there is a connection.

So the question is:

Does it make sence to talk about best approximations to an adjoint functor? If yes, what is the correct definition? Have such approximations been studied? If yes, can you, please, give a reference? What if categories $F$ and $G$ have some extra-structure (say, are DG categories)?

Thank you very much!

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A left adjoint for $G$ is precisely an absolute right Kan extension of $\mathrm{id}$ along $G$. –  Zhen Lin Jan 31 at 8:39
    
There is probably hope that your functor satisfies some weaker condition than admitting a right adjoint, but it probably depends on why you want it to admit an adjoint. –  Adeel Feb 1 at 10:38

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