that $G=UF$ as comonads. Indeed, let $F$ take an object $X$ to the $G$-comodule $(GX,{\overrightarrow}G_2)$. The counit of the adjunction is $\varepsilon: UF=G \to \C$ and the unit $\eta$ evaluated at a $G$-comodule $(V,{\overrightarrow}v)$ is $$\xymatrix{
V\ar[r]^-{1.G_0}&
V.GI\ar[r]^-{{\overrightarrow}v}&
GV.}$$ This is a morphism of $G$-comodules, in the sense of Definition \[def:G\_check\_comodule\], by commutativity of the diagram $$\xymatrix{
V.GX\ar[r]^-{1.G_0.1}\ar[d]^-{1.G_0.1}\ar@/_3pc/@{=}[dd]&
V.GI.GX\ar[r]^-{{\overrightarrow}v.1}\ar[d]^-{1.\delta.1}&
GV.GX\ar[dd]^-{\delta.1}\\
V.GI.GX\ar[r]^-{1.GG_0.1}\ar[d]^-{1.G_2}&
V.G^2I.GX\ar[d]^-{1.G_2}\\
V.GX\ar[r]^-{1.G(G_0.1)}\ar[d]_-{{\overrightarrow}v}&
V.G(GI.X)\ar[d]^-{{\overrightarrow}v}&
G^2V.GX\ar[d]^-{G_2}\\
G(V.X)\ar[r]_-{G(1.G_0.1)}&
G(V.GI.X)\ar[r]_-{G({\overrightarrow}v.1)}&
G(GV.X)}$$ for any object $X$ of $\C$. The region on the right commutes by the fusion equation on ${\overrightarrow}v$; the top-left square commutes by the monoidality of $\delta$, and the regions below it commute by the naturality of $G_2$ and ${\overrightarrow}v$, respectively. The leftmost region commutes by the unitality of the monoidal structure $(G_2,G_0)$. Evaluating $\eta$ at a $G$-comodule of the form $GX=(GX,{\overrightarrow}G_2)$, we obtain $G_2(\delta.1)(1.G_0)=\delta$ (where the equality follows by the functoriality of the monoidal product and the unitality of $(G_2,G_0)$). Hence the first triangle condition on the adjunction $U\dashv F$ follows by the counitality of $\delta$. The other triangle condition holds by the counitality of ${\overrightarrow}v$ and the monoidality of $\varepsilon$. In order to see that the evidently conservative left adjoint functor $U$ is comonadic, we need to prove that it 