This is shown as reference # 115 on page 12 of the Beseler 23CIII manual. It is catalog # 10-43718, Constant Force Spring Assembly. It is listed and named on page 13.
It’s a continuous-band type wind-up spring, something like a Pullman sash balance, but in a different-shaped package. Springs of this type are used for counterbalance force on a variety of enlargers: 8” x 10” Elwood, Omega B66 and D-series, and DeVere to name a few.
I’ve not dealt with the 23CIII. From the looks of it, I speculate that the carriage must be all the way up at the top of the column in order to attach the spring housing to the carriage. Putting the carriage at the top position should leave the spring as relaxed as possible for easy installation.
It looks like you’d first attach the end of the spring to the top of the column at the right, then raise the carriage to attach the spring housing to the carriage. Without seeing it up close and examining the enlarger, spring housing, and associated parts, a guess is the best I can offer.
http://backglass.org/duncan/darkroom/beseler_23ciiixl_manual.pdf
There is contact information on Beseler’s homepage.
https://www.beselerphoto.com/photography-enlargers/