Let $A$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$. Is the canonical functor $$\mathsf{Mod}(A) \to \varprojlim_n ~ \mathsf{Mod}(A/\mathfrak{m}^n),~ M \mapsto (M/\mathfrak{m}^n M)_n$$ an equivalence of categories? If not, does it hold when we restrict to finitely generated modules?
Edit: Meanwhile I've read this many times for finitely generated modules. This seems to be a well-known statement which is related to Grothendieck's existence theorem. Any elementary reference?