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so the correspondence would come from him on behalf of the Committee (although both parties also do their own press releases, etc).

As I noted earlier, these "depositions" are done by the staff and lawyers, not the Committee members per se. You might recall the clips of depositions from several members of the 45 first administration including Jason Miller and Bill Barr, shown by the January 6th Committee (from their "first" big hearing, and you see there is just a handful of people in there with the witnesses and their own lawyers) -

The GOP has wiped most of the J6 Committee stuff although obviously there are copies in other locations, but as an example, here is a copy of a J6 Committee Subpoena to Pat Cipollone, with Bennie Thompson featured on the letterhead - https://www.justsecurity.org/wp-content/uploads/2022/07/january-6-clearinghouse-cipollone-subpoena-letter-june-29-2022.pdf

The issue might be how much of the deposition would they release and whether they would cherry pick out of it (where the (D)s on the Committee will need to vociferously monitor this).