Want to take a content type already in use in many site collections and use it in a content type hub to update the content type in those collections?

I have a custom content type call it "Custom Content Type" which is already in use in many 6 different site collections.  I now have a content type hub.  I want to be able to update the "Custom Content Type" in the content hub and have it propagate to the site collections instead of having to edit the type in the 6 different site collections.  Can I do this or do each of these content types have a different ID even though they have the same name?