Assume $\kappa > \aleph_1$ is regular and let $P=(P_\alpha, \dot{Q}_\beta: \alpha \leq \kappa^+, \beta< \kappa^+)$ be an iteration such that:

1. For even $\beta, \dot{Q}_\beta$ is forced to be $\aleph_1$-closed, $\kappa$-c.c. and a subset of $V_\kappa$.

2. For odd $\beta, \dot{Q}_\beta$ is forced to be $\kappa$-closed,

3. For $p \in P,$ if $Supp(p)$ denotes the support of $p$, then $|Supp(p) \cap E| < \aleph_1$ and $|Supp(p) \cap O|< \kappa$ (where $E$ and $O$ are the class of all even and odd ordinals respectively).

From $P$ we can naturally define the forcing notions $P^E$ and $P^O$, where $$P^E=\{p \in P: \forall \beta \in O, p\restriction \beta \Vdash p(\beta)=1 \}$$ and $$P^O=\{p \in P: \forall \beta \in E, p\restriction \beta \Vdash p(\beta)=1 \}$$. Note that there is a natural map $$\pi: P^E \times P^O \to P$$ which is defined as follows: $\pi(p, q)=r$, where for even $\beta, r(\beta)$ is forced to be $p(\beta)$ and for odd $\beta$ it is forced to be $q(\beta)$.

Question. Is $\pi$ a projection of forcing notions?

Assume $\kappa > \aleph_1$ is regular and let $P=(P_\alpha, \dot{Q}_\beta: \alpha \leq \kappa^+, \beta< \kappa^+)$ be an iteration such that:

1. For even $\beta, \dot{Q}_\beta$ is forced to be $\aleph_1$-closed, $\kappa$-c.c. and a subset of $V_\kappa$.

2. For odd $\beta, \dot{Q}_\beta$ is forced to be $\kappa$-closed,

3. For $p \in P,$ if $Supp(p)$ denotes the support of $p$, then $|Supp(p) \cap E| < \aleph_1$ and $|Supp(p) \cap O|< \kappa$ (where $E$ and $O$ are the class of all even and odd ordinals respectively).

From $P$ we can naturally define the forcing notions $P^E$ and $P^O$, where $$P^E=\{p \in P: \forall \beta \in O, p\restriction \beta \Vdash p(\beta)=1 \}$$ and $$P^O=\{p \in P: \forall \beta \in E, p\restriction \beta \Vdash p(\beta)=1 \}$$. Note that there is a natural map $$\pi: P^E \times P^O \to P$$ which is defined as follows: $\pi(p, q)=r$, where for even $\beta, r(\beta)$ is forced to be $p(\beta)$ and for odd $\beta$ it is forced to be $q(\beta)$.

Clearly $P^O$ is $\kappa$-closed.

Question. Is $\pi$ a projection of forcing notions, assuming $P^E$ is $\kappa$-c.c.?

Remark. I think the extra assumption ``$P^E$ is $\kappa$-c.c.'' is necessary for the question, as otherwise, maybe the arguments from "The $\aleph_2$-Souslin Hypothesis" may be used to show that for some suitable choice of the forcings, $P$ may collapse $\kappa,$ while this can not happen for the product $P^E \times P^O$ by Easton's lemma.

Assume $\kappa > \aleph_1$ is regular and let $P=(P_\alpha, \dot{Q}_\beta: \alpha \leq \kappa^+, \beta< \kappa^+)$ be an iteration such that:

1. For even $\beta, \dot{Q}_\beta$ is forced to be $\aleph_1$-closed and $\kappa$-c.c.

2. For odd $\beta, \dot{Q}_\beta$ is forced to be $\kappa$-closed,

3. For $p \in P,$ if $Supp(p)$ denotes the support of $p$, then $|Supp(p) \cap E| < \aleph_1$ and $|Supp(p) \cap O|< \kappa$ (where $E$ and $O$ are the class of all even and odd ordinals respectively).

From $P$ we can naturally define the forcing notions $P^E$ and $P^O$, where $$P^E=\{p \in P: \forall \beta \in O, p\restriction \beta \Vdash p(\beta)=1 \}$$ and $$P^O=\{p \in P: \forall \beta \in E, p\restriction \beta \Vdash p(\beta)=1 \}$$. Note that there is a natural map $$\pi: P^E \times P^O \to P$$ which is defined as follows: $\pi(p, q)=r$, where for even $\beta, r(\beta)$ is forced to be $p(\beta)$ and for odd $\beta$ it is forced to be $q(\beta)$.

Question. Is $\pi$ a projection of forcing notions?

Assume $\kappa > \aleph_1$ is regular and let $P=(P_\alpha, \dot{Q}_\beta: \alpha \leq \kappa^+, \beta< \kappa^+)$ be an iteration such that:

1. For even $\beta, \dot{Q}_\beta$ is forced to be $\aleph_1$-closed, $\kappa$-c.c. and a subset of $V_\kappa$.

2. For odd $\beta, \dot{Q}_\beta$ is forced to be $\kappa$-closed,

3. For $p \in P,$ if $Supp(p)$ denotes the support of $p$, then $|Supp(p) \cap E| < \aleph_1$ and $|Supp(p) \cap O|< \kappa$ (where $E$ and $O$ are the class of all even and odd ordinals respectively).

From $P$ we can naturally define the forcing notions $P^E$ and $P^O$, where $$P^E=\{p \in P: \forall \beta \in O, p\restriction \beta \Vdash p(\beta)=1 \}$$ and $$P^O=\{p \in P: \forall \beta \in E, p\restriction \beta \Vdash p(\beta)=1 \}$$. Note that there is a natural map $$\pi: P^E \times P^O \to P$$ which is defined as follows: $\pi(p, q)=r$, where for even $\beta, r(\beta)$ is forced to be $p(\beta)$ and for odd $\beta$ it is forced to be $q(\beta)$.

Question. Is $\pi$ a projection of forcing notions?