Just for the record:
You don't even need to have the Remote connected when you update the Profiler. You can keep the Remote unused in a closet for months and skip multiple Profiler updates. Once you connect the Remote, it will automatically be updated.
TLDR: Yes
To explain what may be going on under the hood: most probably, the profiler and the remote do kind of a "handshake" and compare their software versions for compatibility. Again, most probably, the firmware flash file itself is stored in the memory of the profiler and is copied over ethernet to the uC of the remote, where it is flashed. After successful flashing the remote reboots, does the handshake again and the profiler recognizes the remote's firmware to be compatible with the one on the profiler. This means that the profiler kind of "carries" the flash file for the remote within itself until a remote is connected which requires to be flashed. As I said: pure speculation and educated guessing.