U.S. Capitol Police are under the Capitol Sgt at Arms who works for the Speaker. (Note: The Senate also has a Sgt at Arms)
Metropolitan Police Department and Department of Corrections are under Mayor Muriel Bowser.
These are NOT the same entities, but work together. DC is not in any state. The relations between the speaker’s office, federal courts, and metropolitan authorities are murky at best.
I believe that the J6 detainees are Federally charged, which would fall under the U.S. Attorney General Jerk Garland.
Hope that makes things clear as Mud.
(see if you can figure it out better if you want)
https://www.house.gov/the-house-explained/legislative-branch-partners/u-s-capitol-police
https://mpdc.dc.gov/page/mpd-organization
:)