$\omega$ , modulo operation, Boolean algebras

Question

In this paper of Shelah on the page 6 in the -3rd line, what it means $$\bigwedge_{i<\kappa}h(i)=i \text{ mod } \omega$$? I just do not understand the notation.

Answer 1

The only way I can understand this would be that if $i=\delta+n$ for some $n<\omega$ and a limit ordinal $\delta$, then $h(i)=n$. In other words, $i$ and $h(i)$ are the same number of successors from a limit ordinal or $0$.

Of course the $\bigwedge$ is just an infinitary conjunction, so in other words $\forall i<\kappa$, etc.