Copying c++ lambda to Function Pointer reference

I am not sure if I have defined behaviour in the following situation:

My Function pointer type:

typedef void (*DoAfter_cb_type)(void);

The Function which should assign callbacks:

void DoSomething(DoAfter_cb_type & DoAfter_cb)
{
    //...
    DoAfter_cb =  [](){
        //...
    };
}

Caller:

DoAfter_cb_type DoAfter_cb = nullptr;

DoSomething(DoAfter_cb);

// Here is something that has to be done after DoSomething but before DoAfter_cb.

if( DoAfter_cb != nullptr){
    DoAfter_cb();
}

As I learned here lambdas can be implicitly converted to function pointers.

However thoose are still pointers and I fear that something important for calling the lambda is stored on stack and would be out of scope if I just return the function pointer

I have to use function pointers because i do not have access to std::function in my environment. With std::function I would expect the lambda object to be stored in the reference variable and I would not have any problems.

Is the behaviour the same as If I would just define an ordinary function or do I have any side effects here?