Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set $B\subset{}^\omega2\times{}^\omega2$ with code in $M$ such that $A=B_c$.

I have seen this result stated in some texts, but I have not been able to complete a proof. Here, Andreas Blass gave me a hint to prove the analogous fact for null sets in random extensions, but I have failed to adapt it to the present case. Namely, I used there that the set $R(M)$ of random reals over $M$ has outer measure $0$ (although it is non-measurable). Here I would need that the set $C(M)$ of Cohen reals is comeager in $M[c]$, but I suspect this is false. At least, I cannot apply the 0-1 law to $C(M)$ if it does not have the Baire property.