How can I use a custom type as key for a map in C++?

I am trying to assign a custom type as a key for std::map. Here is the type which I am using as key:

struct Foo
{
    Foo(std::string s) : foo_value(s){}

    bool operator<(const Foo& foo1) {   return foo_value < foo1.foo_value;  }

    bool operator>(const Foo& foo1) {   return foo_value > foo1.foo_value;  }
    
    std::string foo_value;
};

When used with std::map, I am getting the following error:

error C2678: binary '<' : no operator found which takes a left-hand operand of type 'const Foo' (or there is no acceptable conversion) c:\program files\microsoft visual studio 8\vc\include\functional 143

If I change the struct to the one below, everything works:

struct Foo
{
    Foo(std::string s) : foo_value(s)   {}

    friend bool operator<(const Foo& foo,const Foo& foo1) { return foo.foo_value < foo1.foo_value;  }

    friend bool operator>(const Foo& foo,const Foo& foo1) { return foo.foo_value > foo1.foo_value;  }
    
    std::string foo_value;
};

Nothing changed, except that the operator is overloaded as friend. Why does my first code not work?