I know it's easier to give advice than follow it but since you did ask, I would replace the coax with CAT-6. Sure, there are adapters that allow Ethernet including POE to run over coax but there's the additional expense plus that adds more components to manage and possibly fail.
Be sure to install a quality CAT-6 that is of solid (not stranded) pure copper (not CCA/Copper Clad Aluminum) with a jacket that conforms to the application, usually CMR (Riser) for in wall, crawl spaces, attic and between floors.
Also, I suggest 23AWG over the standard 25 to accomodate increased POE ability over longer distances, just to add to your "future proofing" intent.
Of course, during installation do not allow kinks in the cable, which can happen especially when using an existing cable to pull in another: I HIGHLY suggest you have someone feed the new cable off the spool or out of the box as you pull with the existing coax in order to rotate/uncoil as needed to prevent the kinks. It's easier to keep the cable flat when un-rolled off a spool that's on a pipe than it is from a coil in a box.....you have to lift and rotate the box to keep the cable flat and not allow tiny coils that become those unwanted kinks.
And lastly, be sure to use the correct RJ-45 terminations for the purchased cable (for solid conductor and for the conductor's gauge), use the right
crimp tool and properly install and test your terminations.
EDITED 4/2/26 @ 1015 CDT: Just to clarify, the link added to "c r i m p t o o l" above in the last line was added by forum software and is NOT my personal recommendation.
