I ran across the following statement in a paper, and it seems fishy to me:

### Lemma: If $A$ is any Hopf algebra, and if $U$ is an $\mathbb{N}_0$-filtered $A$-module algebra, then $U$ and $\mathrm{gr} (U)$ are isomorphic as $A$-modules.

There is no proof of the lemma, it just states that it is a well-known fact.

Such an isomorphism cannot be canonical: consider just the case that $A = k$ is a field and $U$ is any filtered $k$-algebra. In this situation there are plenty of vector space isomorphisms of $U$ with $\mathrm{gr}(U)$, just by pulling back a basis of each $U_{n} / U_{n-1}$, but these are hardly canonical.

So if the lemma is true, it is saying that there is some way to choose one of these maps so that it is an isomorphism of $A$-modules.

### Question

Is the lemma true? If no, what is a counterexample? If yes, could you please provide a proof or a reference?