The motivation for this question is that I am working through an exercise to force the GCH (generalized continuum hypothesis) over a model of ZFC and obtain a model of ZFC where GCH holds.

The forcing is an ORD length iteration of $Add(\gamma^+, 1)$ at every cardinal $\gamma$ ($\gamma$ is a cardinal in $V^{P_{\gamma}})$, with Easton support. I'll just say here what Easton support is:

1) at regular $\gamma$, $P_{\gamma}$ is the direct limit of $P_{\alpha}$, $\alpha < \gamma$, meaning that for all $p \in P_{\gamma}$ there is a $\beta < \gamma$ such that $p\restriction \beta \in P_{\beta}$ and for all $\xi \ge \beta$, $p(\xi) = 1$

2) otherwise, $P_{\gamma}$ is an inverse limit, meaning for all $p \in P_{\gamma}$, the restriction of $p$ to any smaller cardinal is in that stage of the forcing, and there is no imposition that the coordinates of $p$ be trivial past some smaller stage.

Note that at successor cardinals $\gamma$, $P_{\gamma}$ is automatically a direct limit if we force at cardinal stages.

My question is about forcing at singular cardinal stages. These mathoverflow questions, answers, and the surrounding comments (what goes wrong, definition of easton product), were very helpful to me, but I still have a confusion.

Let me first explain what I do understand regarding forcing the GCH at singular cardinals and then my question.

Since $\gamma^+$ is regular for all $\gamma$, the important arguments to show that when we factor $P \cong P^{\le \gamma}*P^{> \gamma}$, the first factor satisfies the $\gamma^+$-chain condition and the second is $<\gamma^+$-closed, all go through, whether $\gamma$ is regular or singlular.

However, part of the above claim rests on a $\Delta$-system argument to show that $P^{\le \gamma}$ satisfies the $\gamma^+$-chain condition. And it is here where we use that any condition $p \in P^{\le \gamma}$ has a small domain compared to $\gamma^+$, and so a $\gamma^+$ sized set of domains of conditions in $P^{\le \gamma}$ forms a $\Delta$-system.

What can I say in the case of singular cardinals? How can the $\gamma^+$ sized set of domains of conditions in $P^{\le \gamma}$ form a $\Delta$-system if we used an inverse limit to support the iteration at that stage? Why not impose a direct limit at every stage of forcing?