Cartier dual is lattice of strict characters

We’ll unpack explicitly stack $\check{G}:=Ma{p}_{CGp}(G,{\mathbb{G}}_m)$ $\check{G}(A)\simeq Ma{p}_{CGp}(G(A),{\mathbb{G}}_m(A))$ Hence, a point $Spec(A)\to \check{G}$ is data of an assignment $\{Spec(A)\text{-points in }G\}\to \{\text{strict units of }A\}$

We can be more explicit here

Map_{Sp_{\ge 0}}(B^\infty G(A), Map_{Sp}&(H\mathbb Z, gl_1(A))) \\ &\simeq Map_{Sp_{\ge 0}}(H\mathbb Z, Map_{Sp}(B^\infty G(A), gl_1(A))) \\ &\simeq Map_{Sp_{\ge 0}}(H\mathbb Z, Map_{CGp}(G(A), A^\times))) \end{align}$$ Hence, $\check G$ is like the stack of strict elements of the "Cartier dual with respect to smooth $\mathbb G_m$", where "smooth $\mathbb G_m$" as a functor computes (not-strict) units.