How to use a function template as a template parameter of another template?

I have some function templates with same signature, and I'm trying to create a proxy function to call them with different types.

How to pass the function name to the proxy function correctly?

https://godbolt.org/z/4vnos3hEf

template <typename T> 
T function()
{
    return T();
}
    
template<template<typename> typename func>
void proxy1(int x)
{
    if (x == 0)
        func<bool>();
    else if (x == 1)
        func<int>();
}

void proxy(int x)
{
    if (x == 0)
        function<bool>();
    else if (x == 1)
        function<int>();
}

int main()
{
    proxy1<function>(0);
    proxy(0);
}

With the proxy() function, there is no problem, but I want to use proxy1() which takes a function as a template parameter.

Compiler complains about:

<source>(26): error C2672: 'proxy1': no matching overloaded function found"

As per Patrick's comment, you need to make your callable object a class template with an operator () member function, something like:

template <typename T> struct call_me
{
    T operator () () { return T (); }
};

template<template<typename> typename callable>
void proxy1(int x)
{
    if (x == 0)
        callable<bool>{}();
    else if (x == 1)
        callable<int>{}();
}

void proxy(int x)
{
    if (x == 0)
        call_me<bool>{}();
    else if (x == 1)
        call_me<int>{}();
}

int main()
{
    proxy1<call_me>(0);
    proxy(0);
}