A Criterion for a morphism to be a counit of an Adjunction 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: 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.

