To begin with, bluntly, Fox is part of the Republican Party. American political parties are made up of both formal organizations (such as the RNC) and informal networks. Fox News Channel, then, is properly understood as part of the expanded Republican Party, just like Hill staff of GOP Members of Congress, or pollsters who only work for Republicans, or activists who volunteer for Republican campaigns, or think tanks that generate legislation for Republicans to support. So, in the first place, Fox is simply part of the communications arm of the party.
The tricky part is that FNC isn’t only a component of the Republican Party. It’s also a business, so it may have profit motivations beyond its partisan goals (both on the organizational and individual level, of course). It’s also, in format, a cable news network, and there are a variety of norms that come with that – norms that may be important to both individual correspondents and producers on the one hand, and consumers on the other. Fox may be part of the communications arm of the Republican Party, but it’s not the same part as the RNC’s Web page, or ads for GOP candidates.
Not sure I totally agree with this. Fox seems to be part of the conservative movement, just like the GOP is now part of the conservative movement.