The part i'm interested in, is the part between the brackets. One side note is that the forcing order is reversed and that's why the disjunction is the greatest lower bound instead of least upper bound. My confusion here is the part which i have underlined in the image. How do we take the conjunction of such $q \Rightarrow p_q(\gamma)$? My first guess is to embed $\mathbb{P}^{\alpha \beta}$ into a complete boolean algebra and go from there. But this idea has an apparent issue and that is the fact that $q \not \in \mathbb{P}^{\alpha \beta}$ and we can't calculate $q \Rightarrow p_q(\gamma)$. What is he doing here? Because the disjunction is crucial in the proofs of his last lemmas, i can't skip it. Any help would be appreciated.
Edit I: One idea i have is is to outright ignore that $\Rightarrow$ and go with this: Let $p(\gamma)$ be a name for the union of the $p_q(\gamma)$ such that the names for the $p_q(\gamma)$ are altered so that when a generic set is chosen containing another $q'$ from the antichain like $G$ then $p_q(\gamma)$ is empty. This does seem to meet the requirements by writing them down. Is this correct? If so, can we extract the meaning of $\Rightarrow$ from this?(Because it appears again in the bottom of the page.)
The point here is that when you iterate, you can amalgamate a bunch of names of conditions in "future iterations" into a single condition in the full length iteration.
Namely, given a bunch of conditions in $\Bbb P_\beta$, whose projections to $\Bbb P_\alpha$ all extend some fixed $q'$, we have a condition in $\Bbb P_\beta$ whose projection to $\Bbb P_\alpha$ is $q'$, and the content of condition in $\Bbb{P_\beta/P_\alpha}$ is decided by some extension of $q'$ in $\Bbb P_\alpha$.
To be more formal, this means that given $q'$ we take a maximal antichain below it in $\Bbb P_\alpha$, and to each $q$ in the antichain we attach a condition $p_q\in\Bbb P_\beta$ such that $p_q\restriction\alpha=q$, and then we define a condition $p$ in $\Bbb P_\beta$ such that $p\restriction\alpha=q'$, and for each $q$ in the antichain, if $q$ is in the generic for $\Bbb P_\alpha$, then $p$ is interpreted the same as $p_q$.
In principle it just means that $\Bbb{P_\beta\cong P_\alpha\ast P_\beta/P_\alpha}$. The key point of Baumgartner is that this amount of freedom does not exist in a product, since in a product we require the conditions of the iteration to be canonical ground model names, rather than arbitrary names which allow for some genericity to be incorporated into the partial order.