Better understanding processing flow around authentication for multiple StackExchange sites

Question

I'm looking at creating a new site that will allow users to interact with multiple Stack Exchange sites and to use their Stack Exchange credentials to join and login to my site. I've been reading the specs on how to do this at http://api.stackexchange.com/docs/authentication but I'm not understanding how to handle multiple Stack Exchange sites. From the viewpoint of authentication do I not worry about which Stack Exchange site a user has and instead just authenticate directly against all Stack Exchange sites via https://stackexchange.com/oauth? Or do I need to go to a different url per site (e.g. https://stackoverlow.com/oauth) or pass some token to uniquely identify which Stack Exchange site I want to authenticate against? nd