Let $\operatorname{Col}(\omega,<\kappa)$ denote the Lévy collapse of an inaccessible cardinal $\kappa$. A variant of the Factor Lemma is as follows:

Lemma.Suppose that $\kappa$ is an inaccessible cardinal and that $\mathbb{P}$ is a poset of size $<\kappa$. Let $G$ be $\operatorname{Col}(\omega,<\kappa$)-generic over $V$. If in $V[G]$ there is a filter $h \subseteq \mathbb{P}$ that is $\mathbb{P}$-generic over $V$, then there is $G^* \in V[G]$ that is $\operatorname{Col}(\omega,<\kappa)$-generic over $V[h]$ and such that $V[h][G^*] = V[G]$.

For those interested, I took this variant from the paper *Happy and mad families in $L(\mathbb{R})$*, Lemma 17 of Page 10.

This lemma shows that, in particular, $h \in V[G]$. Is it true that there must exist some ordinal $\beta < \kappa$ such that $h \in V[G\upharpoonright\beta]$?