$k$ : algebraically closed field

$\mathcal{C}$: category of local Artinian $k$-algebras with residue field $k$

$\hat{\mathcal{C}}$: category of complete local $k$-algebras with residue field $k$

Can you give an example to show that not every covariant functor $F:\hat{\mathcal{C}}\rightarrow (Sets)$ is of the form $\hat{G}$ for some $G:\mathcal{C}\rightarrow (Sets)$?