If a search engine offered you to put your logo and adsense code on a search engine (hosted on there servers) without any ads from the search company, would you use it and place a search box on your website?
As with anything you'd have to look at what benefit they're getting. So far you haven't explained how this can be a win:win If they're doing all the work and you're getting all the benefits then that falls into the "if it seems too good to be true then it probably is" category and you need to ask more questions.