Jackie...

The interesting problem here is whether Morgan horse is an adjective plus noun function or a compound noun function. If Morgan horse doesn't appear to be identified in any dictionary as a noun, then you'd use it as an adjective plus a noun--and it would be diagrammed differently from American saddle horse (all on one line functioning as a noun).

I'll check onelook right now to see whether any dictionary identifies 'Morgan horse' as a noun--and will return if there is one that so lists it.

Edit: Tough call. Three more encyclopedia-like references do list 'Morgan horse' on onelook--and that's out of over 800 references, including MW and AHD--but none of the three name 'Morgan horse' as a noun, as is true of other breeds of horses we've discussed above. Even though it's not recognized by the standard dictionaries as a compound word that functions as a noun--doesn't it follow suit from other horse breeds that function as nouns? I figure if Shetland pony, American saddle horse, Arabian horse all function as nouns, then Morgan horse should, too, whether it appears as such in a dictionary. Otherwise, it would just be too exceptional of an exception.