Transferring external iTune library from one computer to another

Question

I am having great difficulty in changing my external iTunes library from my old MBP laptop to my new iMac 5k.
My entire iTunes library is stored on an external drive and can be accessed directly with my MBP. When I started up iTunes on my new iMac, it created a new empty library in the MUSIC folder. What I want to do is connect my new iMac to the external library directly. I can see it when I turn on HOME SHARING but only when iTunes is running on my MBP, which is useless since my MBP will now be seldom used. I have of course on the iMac, changed the library location in the advanced preferences to the external location, to no avail. I do not want to keep any media files on the iMac for capacity reasons.
To compound the problem, I cannot seem to sync my new iPhone 6+ to the original external library, only the new empty one!
Surely this ought to be a simple job, transferring an external iTunes library from one computer to another! Am I missing something obvious here?

Answer 1

Wow, what a quick response! Thanks, Limnos. I guess it was bleeding' obvious, and I should have thought of it before. Such a simple solution!
On a related matter, I also have just realised that the iTunes iOS backups are saved on the internal computer drive in a folder called MOBILESYNC rather than on an external drive, thus chewing up a lot of capacity on my SSD. Others reading this response may be interested to know that it is easy to have iTunes save the iOS backups to any external drive using a symbolic link in this directory. Go to
http://support.digidna.net/hc/en-us/articles/203504123-Storing-your-iPhone-Backu ps-on-an-Alternate-Location
for the details.
However, note that iBooks will NOT let you use this work-around, pathetic app that it is.