Let $(A_n)<$ be a projective and inductive system of $\mathbb{Z}/p^n$-modules.
Is then $\operatorname{Hom}{(\projlim A_n, \mathbb{Z}_p)}$ isomorphic to $\operatorname{Hom}(\operatorname{Div}(\injlim A_n), \mathbb{Q}_p/\mathbb{Z}_p)$, resp. under which additional (finiteness?) assumptions does this hold? $\operatorname{Div}$ denotes the maximal divisible subgroup.