What will be mandatory is that the COM port to which the remote is attached be the FIRST available and unused COM port. That is, if mouse is on COM1, and COM2 is available, remote can be connected there. However, if COM1 is unused and your adapter is configured for COM2, connecting remote there will not work and the adapter should be set for use as COM1.
Many adapters provide advanced driver settings to allow one to specify such a thing. If your adapter does not have this then, I suggest the Keyspan as documented in the FAQ or possibly the use of Tonto from [Link: giantlaser.com]