Do I get the alternator core charge back or does the merchant keep it ? on 2005 Mercury Mariner

I was charged 65.00 towing, 100.00 alternator assembly, 287.00 alternator, 165.00 labor.and they refused to return the old alternator so I could get my core charge back.
Is this the correct way for a merchant to handle this ?

Asked by for the 2005 Mercury Mariner
On a new alternator there is no core charge, new alternators are expensive from Mercury, some Chinese alternators are available new. On a re manufactured alternator there is a core charge they need to have something to rebuild, you can get yours back if you forfeit the core charge.
1 more answer
Just a note: Ford does not sell anything except for remanufactured parts, and they all have core charges associated with them.