How I can use mongoose and Koa.js

Wrap your database initialization into a Promise:

const Cat = mongoose.model('Cat', { name: String });

function getCats() {
  return new Promise((resolve, reject) => {
     const ourCat = Cat.find({ name: 'Zildjian' });
     ourCat.exec((er, cats) => {
       if (er) {  reject(er);   }
       else { resolve(cats); }
     });        
  });
}

So then you can just do:

const connection = connectDB(mongoUri);
app.use(async ctx => {
   await connection;
   ctx.body = await getCats();
});