The classical Cantor-Schroder-Bernstein Theorem says that there exists a bijective function $X\leftrightarrow Y$ if and only if there exist injective functions $X\hookrightarrow Y$ and $Y\hookrightarrow X$.
It is known that without Axiom of Choice the injectivity cannot be replaced by the surjectivity in this result.
Nonetheless, without AC, we have the following simple
Fact. If for two sets $X,Y$ there are surjective functions $X\twoheadrightarrow Y$ and $Y\twoheadrightarrow X$, then there exists a bijective function $\mathcal P(X)\leftrightarrow \mathcal P(Y)$ between the corresponding power-sets.
Can the power-sets in this result be replaced by some simpler sets built over $X$, for example, $X^\omega$ or $X\times\omega$?
Question. Assume that for two sets $X,Y$ there exist surjective functions $X\twoheadrightarrow Y$ and $Y\twoheadrightarrow X$.
Is it true (in ZF) that the sets
(i) $X^\omega$ and $Y^\omega$ have the same cardinality?
(ii) $X\times\omega$ and $Y\times\omega$ have the same cardinality?