Suppose we have two functors $F:C\leftrightarrow D:G$ and a morphism $\varepsilon:FG\rightarrow\operatorname{Id}\_D$. I am looking for a way to check whether $\varepsilon$ is the counit of an adjunction. Mostly I am interested in a way to show that $\varepsilon$ is *not* a counit, i.e. necessary conditions.



A Criterion for a morphism to be a counit of an Adjunction
A morphism $\varepsilon\colon FG\to Id\_D$ is the counit of adjunction $F\dashv G$ iff for every $d\in D$ the morphism $\varepsilon\_d\colon FGd\to d$ is universal from $F$ to $d$(i.e. is the terminal object of the comma category $(F\downarrow d)$). So we have a family of necessary conditions, parametrized by objects of $D$.


Also, there are interesting connections between $G$ and $\varepsilon$(if $\varepsilon$ is the counit):


* $G$ is faithful iff for any $d\in D$ the morphism $\varepsilon\_d$ is an epimorphism;
* $G$ is full iff for any $d\in D$ the morphism $\varepsilon\_d$ is a split monomorphism.


So, for example, if $G$ is full, and for some $d\_0\in D$ the morphism $\varepsilon\_{d\_0}\colon FGd\_0\to d\_0$ does not have a left inverse, then $\varepsilon$ is not a counit.


Finally, another interesting property you may find useful:


* If one of the functors $F$, $G$ is full, then the natural transformation $G\varepsilon$ is invertible.


References:



> 
> Maclane, CFWM, chapter IV.
> 
> 
> 


**Edit:** As you said, your $\varepsilon$ is invertible. Assuming $\varepsilon$ is the counit, you can also define the unit of the corresponding adjunction on the image of $G$ on objects(it follows from the *triangular identities*):
$$
\eta\_{G(d)}=(G(\varepsilon\_d))^{-1}
$$
Maybe it will help you to find the whole unit morphism. For example, if $G$ is surjective on objects, you don't need to check anything(in this case $(F,G)$ is an equivalence).


Also you can use some additional data about your categories to find the adjunction. For instance, if $D$ is locally small, complete, well-powered, has a small cogenerating family, and $G$ is continuous, then you can find the left adjoint by the *Special Adjoint Functor Theorem*. Then you can try to prove that $F$ is naturally isomorphic to this left adjoint etc.


References:



> 
> Borceux, "Handbook of categorical algebra", vol.1.
> 
> 
> 


