Mac Mail and "Syncronize" function

I like my email accounts synchronized. When I hit refresh in my Hub on my BB10 device the emails will disappear in Apple Mail if I had recently deleted them on my BB10 device. But when I do the "Synchronize all accounts" in Apple Mail if I had recently deleted them there they will not disappear from my BB10 device. Does anyone know how to get this synchronize to work?

Does it work correctly with GMail and or Yahoo?

Originally Posted by MobileMadness002

Does it work correctly with GMail and or Yahoo?

I don't use those.

Also, I'm talking about the Apple Mail email client software on my Mac. Not an online service.