If the business is new business from a new client, as in your example, then the reinsurer will indeed carry out some form of underwriting to assess the risk of the business. They do this not only to decide whether or not to take out the business, but also to help to decide the level of fees/premiums to charge the insurer.
The situation is slightly different if the business is a new policy from an existing client with whom the reinsurer has a treaty agreement. In this case, the insurer and reinsurer would have decided on acceptable risk levels when they started their relationship and this would be stated in the treaty. If any new business which the insurer decides to launch falls within this risk level, then the reinsurer will take it on without underwriting. If the new business, however, is very large/unique/uncertain, therefore, posing a risk greater than the accepted level, the reinsurer would carry out underwriting for this business.
I is also important to remember that the reinsurer will also be monitoring the claims and carrying out their own form of claim stage underwriting to ensure that all the terms of the reinsurance agreement are being met.
I hope this clarifies things.