  Let $G= {\rm Spec} A$ be an affine $R$-monoid scheme. We prove that the
category of dual functors (over the category of commutative $R$-algebras) of
$G$-modules is equivalent to the category of dual functors of ${\mathcal
A}^*$-modules. We prove that $G$ is invariant exact if and only if $A^*= R
\times B^*$ as $R$-algebras and the first projection $A^* \to R$ is the unit of
$A$. If $\mathbb M$ is a dual functor of $G$-modules and $w_G := (1,0) \in R
\times B^* = A^*$, we prove that $\mathbb M^G = w_G \cdot \mathbb M$ and
$\mathbb F = w_G \cdot \mathbb M \oplus (1-w_G) \cdot \mathbb M$; hence, the
Reynolds operator can defined on $\mathcal M$.
