iPhone CONTACTS set to Fetch?

Mail can arrive on your device in a few different ways...

Manual: only get new mail as and when you ask for it to happen, e.g. Dragging down your inbox to kick things off.
Fetch: every so often, in the background. You decide how often.
Push: mail gets pushed immediately onto your device by the sender. iCloud supports this, and it's probably what you want.

Until recently, Apple were banned from pushing iCloud Mail in Germany because of a legal spat. Now, the lucky Germans can have push if they want it.

 

What does mail have to do with contacts?

 

Beats me. The screengrab is to do with mail setup.

 

But it shows an account with just contacts, and the question seems to be more or less as to what contacts is doing there and what would the fetch or other similar settings be doing for it.

 

Depending on how the email server is set up and handles the information, sometimes contacts, emails, and calendar shown all under one setup (such as exchange and iCloud) or it gets broken apart during rage setup (some of my IMAP accounts that handle all the information broken them into separate sections (one had just contacts, one had contacts and calendar for example). I would bet it was one of your other accounts set up on the device that you referred to as existing on it.