Suppose for simplicity we are working over $\mathbb{F}_p$. Cartier duality is an antiequivalence between formal groups and affine group schemes over $Spec(\mathbb{F}_p)$. Let $\mathbb{W}_p(-)$ denote the Witt vector affine group scheme. It is well known that the Cartier dual $Map_{Grp}(\mathbb{W}, \mathbb{G}_m)$ to this is the Witt vector formal group $\widehat{\mathbb{W}_p}$. This can be defined by taking the colimit of all formal completions at the identity of the truncated Witt schemes and comes endowed with natural formal group structure.

I know that Cartier duality should be viewed as exchanging the Frobenius and Verschiebung maps. Does this mean that the Cartier dual of the natural Frobenius map $F: \mathbb{W}_p \to \mathbb{W}_p$ will be the Verschiebung map defined on the Witt vector formal scheme? This seems to be incompatible with the statement that the Cartier dual of the formal multiplicative group $\widehat{\mathbb{G}_m}$ is the subgroup scheme of "fixed points" of $F: \mathbb{W}_p(-) \to \mathbb{W}_p(-)$.