**Notation:**

$M[{\mathbb{P}}:G]$ denotes the forcing extension of $M$ by $\mathbb{P}$-generic filter $G$.

$\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ denotes the ultraproduct of models using the ultrafilter $\mathcal{F}$ on the index set $I$.

Consider the index set $I$, the ultrafilter $\mathcal{F}$ on it, the family $\{M_i~|~i\in I\}$ of countable transitive models of $ZFC$, the collection $\{\mathbb{P}_i~|~i\in I\}$ of forcing notions such that $\forall i\in I~~~\mathbb{P}_{i}\in M_i$ and the collection $\{\mathbb{G}_i~|~i\in I\}$ of sets such that $\forall i\in I~~~G_i$ is a $\mathbb{P}_{i}$-generic filter over $M_i$.

For all $i\in I$, $M_i$ and $M_i[\mathbb{P_i}:G_i]$ are models of $ZFC$.

Also $\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ and $\prod_{\mathcal{F}}\langle M_i[\mathbb{P}_i:G_i]~|~i\in I\rangle$ are models of $ZFC$.

**Question 1:** Is there any natural (categorical) relation between the ultraproduct of forcing extensions of some ground models $\prod_{\mathcal{F}}\langle M_i[\mathbb{P}_i:G_i]~|~i\in I\rangle$ and the forcing extension of a ground model produced by ultraproduct of component ground models $\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$?

For example, is there a forcing notion $\mathbb{P}$ and a $\mathbb{P}$-generic filter $G$ over $\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ such that:

$$(\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle)[\mathbb{P}:G]\cong \prod_{\mathcal{F}}\langle M_i[\mathbb{P}_i:G_i]~|~i\in I\rangle$$

**Question 2:** Let $M$ be a model of $ZFC$ and $I, \mathcal{F}\in M$ and ($\mathcal{F}$ is an ultrafilter on $I$)$^M$ and $\{\mathbb{P}_i~|~i\in I\}$ is a family of forcing notions such that $\forall i\in I~~~\mathbb{P}_i\in M$ and $\{G_i~|~i\in I\}$ a family of sets such that $\forall i\in I~~~G_i$ is a $\mathbb{P}_i$ - generic filter over $M$.

Note that $\prod_{\mathcal{F}}\langle \mathbb{P}_i~|~i\in I\rangle$ is a forcing notion in $M$ also $\prod_{\mathcal{F}}\langle G_i~|~i\in I\rangle\subseteq \prod_{\mathcal{F}}\langle \mathbb{P}_i~|~i\in I\rangle$.

**(a)** Under what conditions the subset $\prod_{\mathcal{F}}\langle G_i~|~i\in I\rangle$ is a $\prod_{\mathcal{F}}\langle \mathbb{P}_i~|~i\in I\rangle$-generic filter over $M$?

**(b)** If $\prod_{\mathcal{F}}\langle G_i~|~i\in I\rangle$ is a $\prod_{\mathcal{F}}\langle \mathbb{P}_i~|~i\in I\rangle$-generic filter over $M$, consider the generic extension $M[\prod_{\mathcal{F}}\langle \mathbb{P}_i~|~i\in I\rangle:\prod_{\mathcal{F}}\langle G_i~|~i\in I\rangle]$, is there a natural relation between this generic extension of ground model and generic extensions $\{M[\mathbb{P}_i:G_i]~|~i\in I\}$?

For example, is $\prod_{\mathcal{F}}\langle M[\mathbb{P}_i:G_i]~|~i\in I\rangle$ isomorphic to some ultrapower of $M[\prod_{\mathcal{F}}\langle \mathbb{P}_i~|~i\in I\rangle:\prod_{\mathcal{F}}\langle G_i~|~i\in I\rangle]$? i.e. is there an index set $J$ and an ultrafilter $\mathcal{U}$ on $J$, such that:

$$\prod_{\mathcal{U}}\langle M[\prod_{\mathcal{F}}\langle \mathbb{P}_i~|~i\in I\rangle:\prod_{\mathcal{F}}\langle G_i~|~i\in I\rangle]~|~j\in J\rangle\cong \prod_{\mathcal{F}}\langle M[\mathbb{P}_i:G_i]~|~i\in I\rangle$$

In the other words, is ultraproduct of a family of generic extensions of a given ground model isomorphic to an ultrapower of generic extension of ground model using ultraproduct forcing notion?

**Remark:** Forcing is a Boolean-valued ultraproduct itself. It seems reasonable to have some sort of natural interactions between ultraproducts.