Moving a catalog (only) onto a new PC while leaving the original image files on an external drive

I have tried to answer this question for myself by looking at a previous thread, however this got very long and complicated and I think with multiple catalogs involved has just left me a little confused.

Would realy appreciate some advice please on the following.  I store all of my image files on an external hard disk (and ensure this and the catalog gets periodically backed up onto separate drives in case I lose the original drive for any reason).  So for this exercise I just want Elements 2025 (already loaded onto both old and new PC) to have the current catalogue on the new PC.  I then just plug the external drive into the new PC and in theory keep everything joined up and the database intact (I realise the external drive must be mapped with the same letter when connected).

I could do a full backup (catlog and images), restore this onto the new pc and then move all the image files back onto the external hard disk where they started from, but this does seem rather cumbersome and time consuming.

Would I be safe to just copy the necessary catalog file (I have completed a catalogue only backup) in explorer onto my new computer (ensuring the external hard disk has the same mapping letter when plugged into the new computer)?  

Depending on answers any info on correct file paths and folder would help.

Many thanks  Jonathan