Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable partial functions from $\kappa$ to 2, ordered by $\tau \leq \sigma$ iff $1 \Vdash \tau \leq \sigma$. Call this $T(\mathbb P, Add(\kappa))$.

It is well known that the identity map is a projection from $\mathbb P \times T(\mathbb P, Add(\kappa))$ to the iteration $\mathbb P * \dot Add(\kappa)$. Furthermore in this situation it is not hard to show that $T(\mathbb P, Add(\kappa)) \cong Add(\kappa)$. So if $G \times H$ is $\mathbb P \times Add(\kappa)$-generic, then in the extension there is $I$ such that $G * \dot I$ is $\mathbb P * \dot Add(\kappa)$-generic.

My question is what about the opposite. In $V[G* \dot I]$ is there an $Add(\kappa)^V$-generic?