I'm going to make some assumptions about injected properties of the controller class. For full setups, you can view my testing example repo here. This should be essentially what you're looking for though

describe('UserController (Unit)', () => {
  let controller: UserController;
  let service: MockedClass<UserService>;

  beforeAll(async () => {
    const modRef = await Test.createTestingModule({
      controllers: [UserController],
      providers: [
        {
          provide: UserService,
          useValue: {
            update: jest.fn().mockReturnValue({ hello: 'world' }),
          },
        }
      ],
    }).compile();
    controller = modRef.get(UserController);
    service = modRef.get(UserService);
  });
  it('should call update and return the value', () => {
    const bodyVal = objectThatMatchesUpdateUserDto;
    const reqVal = {
      user: {
        user: valueThatMatchesReqUserUser
      }
    };
    const res = controller.update(bodyVal, reqVal);
    expect(res).toEqual({ hello: 'world' });
    expect(service.update).toBeCalledWith(reqVal.user.user, bodyVal)
  });
});