I am studying Kunen's set theory book (the new version), and I am thinking about Exercise IV.4.8. This is a problem about dense embeddings between forcing posets. The relevant terminology can be found in Kunen's book. $Fn(I,J)$ means the set of finite partial functions from $I$ to $J$ ordered by reverse inclusion. The finite support product of $|I|$ copies of $\omega^{<\omega}$ means the functions from a set of size $|I|$ to $\omega^{<\omega}$ with only finitely many nonempty values, ordered by pointwise extension.
The exercise asks us to obtain a dense embedding from the finite support product of $|I|$ copies of $\omega^{<\omega}$ to $Fn(I,\omega)$, and another dense embedding from the same finite support product to $Fn(I,2)$.
I would like to verify if my line of thought works, and the following is my attempt.
From Exercise III.3.70 I know that $\omega^{<\omega}$ densely embedds into any countable atomless forcing poset. Given a dense embedding $k:\omega^{<\omega} \rightarrow Fn( \omega, \omega)$, we define a function $j$ from the finite support product of $|I|$ copies of $\omega^{<\omega}$ to $Fn(I \times \omega, \omega)$ as follows: for any element $f$ of the product, define $j(f)$ to be the finite partial functions such that $j(f)(i,m)=n$ iff $f(i)$ is nonempty and that $k(f)(m)=n$. $j(f)$ is undefined othwewise. This way the density of $k$ would ensure that $j$ is also a dense embedding. Now since $Fn( \omega, \omega)$ and $Fn(I \times \omega, \omega)$ are isomorphic, we get a dense embedding from the finite support product to $Fn( \omega, \omega)$.
For $Fn(I,2)$ we similarly appeal to some dense embedding $k:\omega^{<\omega} \rightarrow Fn( \omega, 2)$ to obtain some embedding from the product to $Fn(I \times \omega, 2)$, and then appeal to isomorphism between $Fn(I \times \omega, 2)$ and $Fn(I,2)$.
Is the above argument correct? Are there other solutions?
Thanks!