creates $U$-absolute equalizers. Suppose then that $f$ and $g$ are morphisms $(V,{\overrightarrow}{v})\to(W,{\overrightarrow}{w})$ and that $$\xymatrix{
Z \ar[r]^{h} & V \ar@<.3ex>[r]^{f} \ar@<-.3ex>[r]_{g} & W }$$ is an absolute equalizer in $\C$. Then in the solid part of the diagram $$\xymatrix{
Z.GX \ar[r]^{h.1} 
\ar@{-->}[d]_{{\overrightarrow}{z}} & 
V.GX \ar@<.3ex>[r]^{f.1} \ar@<-.3ex>[r]_{g.1} 
\ar[d]_{{\overrightarrow}{v}} & W.GX \ar[d]^{{\overrightarrow}{w}} \\
G(Z.X) \ar[r]_{G(h.1)} & 
G(V.X) \ar@<.3ex>[r]^{G(f.1)} \ar@<-.3ex>[r]_{G(g.1)} & 
G(W.X)}$$ the rows are equalizers, and the parallel pairs commute serially with the verticals, thus there is a unique induced morphism ${\overrightarrow}{z}$ as in the dashed part of the diagram. The axioms for $(Z,{\overrightarrow}{z})$ to be a comodule follow easily from the corresponding axioms for $(V,{\overrightarrow}{v})$ and the fact that $h$ is an absolute monomorphism. The morphism $h$ preserves the comodule structure by construction, and the universal property of $(Z,{\overrightarrow}{z})$ follows from the universal property of $Z$ along with the fact that $G(h.1)$ is a monomorphism.

Explicitly, the inverse of the comparison functor – from the category of $G$-comodules in Definition \[def:G\_check\_comodule\] to the category of Eilenberg-Moore $G$-comodules – sends an Eilenberg-Moore comodule $v:V\to GV$ to the equalizer of the comodule morphisms $\delta,Gv:(GV,{\overrightarrow}G_2) \to
(G^2V,{\overrightarrow}G_2)$; that is, to the comodule $$\xymatrix{
V.G(-) \ar[r]^-{v.1} &
GV.G(-) \ar[r]^-{G_2} &
G(V.-).}$$

Further examples of the 