I am trying to understand the remark on page 20 of these notes. For $G=\operatorname{Spec} A$ a group scheme, Cartier duality is defined there as $G^\vee(B) = \operatorname{Hom}_{B-grp}(G_B, (\mathbb{G}_m)_B)$.
On the other hand, in the remark it is defined as the sheaf hom $\underline{\operatorname{Hom}}(G, \mathbb{G}_m)$. Using this definition, it seems to me that applied to $B$, it would give
$\underline{\operatorname{Hom}}(G, \mathbb{G}_m)(B) = \underline{\operatorname{Hom}}(G(B), \mathbb{G}_m(B))$,
which does not seem like the same thing. For instance, if $G=\mathbb{G}_m$, the former appears to be given by $\operatorname{Hom}(B[x, x^{-1}], B[x, x^{-1}])$ while the latter appears to be given by $\operatorname{Hom}(B^\times, B^\times)$.
I am probably overlooking something simple so I'd appreciate it if someone could point out what I'm missing.