Should I extend std::less for a comparison functor?

I want to create a shared_ptr content-comparison functor to stand in for std::less in associative containers and std algorithms. I've seen sevearl examples of custom comparators that use the following (or similar) model:

template <typename T>
struct SharedPtrContentsLess {
  bool operator()(const boost::shared_ptr<T>& lhs, 
      const boost::shared_ptr<T> rhs) const {
    return std::less<T>(*lhs, *rhs);
    //or: return (*lhs) < (*rhs);
  }
  //defining these here instead of using std::binary_functor (C++11 deprecated)
  typedef boost::shared_ptr<T> first_argument_type;
  typedef boost::shared_ptr<T> second_argument_type;
  typedef bool return_type;
};

But why wouldn't I want to instead extend std::less? Like so:

template <typename T>
struct SharedPtrContentsLess : public std::less< boost:shared_ptr<T> > {
  bool operator()(const boost::shared_ptr<T>& lhs, 
      const boost::shared_ptr<T> rhs) const {
    return std::less<T>(*lhs, *rhs);
  }
};

Does this buy me anything at all?

I would think this gets me the typedefs for free, as though I was extending the deprecated std::binary_function. In C++03, I actually would be extending it through std::less. However, this would also be portable from C++03 to C++11/14 and even C++17 when std::binary_function will be removed, as it just follows the changes in std::less.

I've read a bunch of answers on StackOverflow regarding std::less use, custom comparison functors, and even some of the Standard specs and proposals. I see specializations of std::less and guidance not to extend STL containers, but I can't seem to find any examples of extending std::less or guidance against it. Am I missing an obvious reason not to do this?