Sort std::vector

• I want to use std::sort() on a std::vector<std::string *>.
• For this, I want to overload <.
• I guess the std::sort() function doesn't call my operator because the vector actually has std::string * as type instead of std::string. But I'm not sure how to specify this.
• I would like to achieve this without using a third parameter in std::sort()

#include <iostream>
#include <vector>
#include <algorithm>

bool operator<(std::string& str_1, std::string& str_2){
    std::cout<<str_1[0] <<" < " << str_2[0] << std::endl;
    return str_1[0] < str_2[0];
}
int main(int argc, const char * argv[]) {
    std::string str1 = "abc";
    std::string str2 = "def";
    std::string str3 = "ghi";
    std::string str4 = "bcd";
    std::string str5 = "fgh";

    //test the operator first
    std::cout << (str1<str2) << std::endl;

    std::vector<std::string *> str_vector;
    str_vector.push_back(&str1);
    str_vector.push_back(&str2);
    str_vector.push_back(&str3);
    str_vector.push_back(&str4);
    str_vector.push_back(&str5);

    std::sort(str_vector.begin(), str_vector.end());

    for (std::vector<std::string *>::iterator it=str_vector.begin(); it!=str_vector.end(); ++it)
        std::cout << ' ' << **it << std::endl;

    return 0;
}

The operator seems to be well implemented, but is never called during std::sort() This is the console output:

a < d
1
 fgh
 bcd
 ghi
 def
 abc
Program ended with exit code: 0

THANKS!