Let $\kappa$ be an uncountable regular cardinal, and suppose that $\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\alpha:\alpha<\delta\rangle$ is a $\kappa$-support iteration of $<\kappa$-strategically closed notions of forcing, i.e.,

a) $\Vdash_{\mathbb{P}_\alpha}``\dot{\mathbb{Q}}_\alpha\text{ is $<\kappa$-strategically complete''}$, and

b) inverse limits are taken at every limit stage of cofinality $\leq\kappa$, and direct limits elsewhere.

It is well-known (even folklore) that the limit forcing $\mathbb{P}_\delta$ is also $<\kappa$-strategically closed (see, e.g., Proposition 7.9 of Cummings chapter in the Handbook of Set Theory).

Is there a place where a proof of this appears in the literature?

I am asking because I need to use that the obvious strategy has some additional properties, and if I can find an appropriate reference then I can avoid taking a rather technical detour in the middle of another proof.