How to use these two class decorators together with functools.update_wrapper?

I was reading about decorators and tried to mix these two examples and make them class decorators instead of regular functions. The first one only lets you run a function once per argument and the second one counts how many times you've run that function. They both work fine separated but when I try to decorate a simple function with both at the same time it fails... Or doesn't really fail but prints an unexpected wrong result. I did some reading and found that the functools module can help but I'm not sure how.

from functools import update_wrapper

class Memoize:
    def __init__(self, func):
        self.func = func
        self.memo = dict()
        update_wrapper(self, func)

    def __call__(self, *args):
        if args not in self.memo:
            self.memo[args] = self.func(args)
        else:
            print("cls decorator. You have printed this before")
        return self.memo[args]


class CallCounter:
    def __init__(self, func):
        self.func = func
        self.calls = 0
        self.__name__ = func.__name__
        update_wrapper(self, func)

    def __call__(self, *args, **kwargs):
        self.calls += 1
        return self.func(*args, **kwargs)


@Memoize
@CallCounter
def doubleprint(x):
    for elem in x:
        print(elem + " " + elem)


doubleprint('Hello')
doubleprint('Hello')
doubleprint('Hello')
doubleprint('Hello')
doubleprint('Bye')

print(doubleprint.calls)

doubleprint('Hello')

doubleprint('Hello')
doubleprint('Hello')
doubleprint('Hello')
doubleprint('Bye')

print(doubleprint.calls)