I'm teaching myself bits and pieces of forcing at the moment, for the purposes of translating them into sheaf-theoretic versions. I'm trying to write down what I feel is a cleaner description of the Easton product of forcing posets, by which I mean a global description rather than one in terms of elements like $$ \left|{(\kappa,α,β) ∈ dom(p) : \kappa\leq \lambda}\right| \lt \lambda \qquad (1) $$ for $p$ in the Easton product $\prod^E P(\kappa)$, which is not very useful at the level of the category of sets if it is not the category of ZF(C)-sets.

I'm confident I understand what this condition does when you do the forcing, namely I think it puts a bound on how many sets are added to any given $\lambda$, else you might get silly things like a proper class of sets added to some set.

However, at the risk of embarrassing myself, and in the interest of educating others, here is my best guess for what this condition translates to. Consider, as Jech does (Set theory, 3rd edition), first the Easton product over some set $A$ of regular cardinals. For $\kappa\in A$, let $P(\kappa)$ be the set of functions $p:D(p)\to 2$ where $D(p)\subset \kappa\times B(\kappa)$ has cardinality less than $\kappa$. Here $B(\kappa)$ is some cardinal, not necessarily given by an Easton function on regular cardinals, and forcing using the 'usual order' on this poset will add $B(\kappa)$ subsets to each $\kappa\in A$. There is a distinguished element $\top$ of each $P(\kappa)$, namely the unique function $\emptyset \to 2$.

Let $P=\prod P(\kappa)$ be the product over $\kappa\in A$. The Easton product is the subset consisting of those $p\in P$ such that the support condition (1) holds. So what does this mean? An element $p\in P$ is a collection of functions $p_\kappa$, one for each $\kappa\in A$. Then let $supp(p) \subset A$ be the set of those $\kappa\in A$ such that $p_\kappa\not=\top$. Analogously, define $supp_\lambda(p) \subset A$ to be the set of those $\kappa\leq \lambda$ such that $p_\kappa\not=\top$. This last definition is where I am the most unsure, as the definition in Jech really involves $supp(p)\cap \lambda$, which doesn't make sense from a structural set theory point of view unless $\lambda$ is viewed as a subset of $A$ (even though it makes perfect sense in a material set theory such as ZFC).

Now assuming the definition of $supp_\lambda(p)$ is correct, the support condition on $p$ as first stated in Jech is that $$ \forall \text{ regular } \lambda, \left|supp_\lambda(p)\right| \lt \lambda.\qquad (2) $$ Apparently it's enough to enforce (2) whenever $\lambda$ is weakly inaccessible, (though Friedman just says 'inaccessible'), and that's fine by me. However, this doesn't look the same as the condition (1), which has me slightly worried. The condition (1) imposes a condition on the size of the domain of $p$ as well as (2), and in fact implies that the domain of $p_\kappa$ is smaller than $\kappa$ for all $\kappa$.

So where has my reasoning gone wrong? At this point I really just want to understand the set underlying the Easton product, rather than any intricacies of class forcing.