How has the un increased the cooperation between countries?

The UN, United Nations is an organization that unites nations, so its very purpose and all its actions are directed at this goal.

One example are the different initiatives and committees that the UN has that span across several countries, or even all or almost all of them. An example is UNESCO, the United Nations organization that protects world-wide  cultural heritage. It's a bonus for tourism to have the country's attractions on UNESCO World Heritage List, and some of those sites are shared, such as Prehistoric pile dwellings around the Alps, which is shared among 6 countries.

Additionally UN works towards combating the aspect that threatens the cooperation most: conflicts and open wars. Specifically, it sends troops to affected regions and negotiates peace talks.